Subjective Game Structures: A behavioral game theoretic analysis of hidden perceptions and strategic properties underlying the Israeli-Palestinian conflict

Fischer, Ilan; Avrashi, Shacked; Givon, Lior Article Subjective Game Structures: A behavioral game theoretic analysis of hidden perceptions and strategic properties underlying the IsraeliPalestinian conflict Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Fischer, Ilan; Avrashi, Shacked; Givon, Lior (2025) : Subjective Game Structures: A behavioral game theoretic analysis of hidden perceptions and strategic properties underlying the Israeli-Palestinian conflict, Games, ISSN 2073-4336, MDPI, Basel, Vol. 16, Iss. 1, pp. 1-23, https://doi.org/10.3390/g16010003 This Version is available at: https://hdl.handle.net/10419/330117 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Academic Editor: Ulrich Berger Received: 27 November 2024 Revised: 21 December 2024 Accepted: 2 January 2025 Published: 7 January 2025 Citation: Fischer, I., Avrashi, S., & Givon, L. (2025). Subjective Game Structures: A Behavioral Game Theoretic Analysis of Hidden Perceptions and Strategic Properties Underlying the Israeli–Palestinian Conflict. Games,16(1), 3. https:// doi.org/10.3390/g16010003 Copyright: © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/). games Article Subjective Game Structures: A Behavioral Game Theoretic Analysis of Hidden Perceptions and Strategic Properties Underlying the Israeli–Palestinian Conflict Ilan Fischer * , Shacked Avrashi and Lior Givon Department of Psychology, Faculty of Social Sciences, University of Haifa, Haifa 3498838, Israel; [email protected] (S.A.); [email protected] (L.G.) *Correspondence: [email protected] Abstract: Here, we apply a novel framework, termed Subjective Game Structures (SGSs), for uncovering and analyzing hidden motivations in ecological conflicts. SGSs enable the examination of implicit attitudes and motivations within individuals and groups. We elicited SGSs from Israeli and Palestinian participants between March 2019 and February 2020 (approximately three years before 7 October 2023), trying to answer the questions of whether Israelis and Palestinians perceived the conflict in a similar manner, whether they have identical assessments of the associated payoffs, and what can be done to reduce future hostilities and attain peaceful solutions. The results reveal meaningful differences between the parties. Israeli SGSs largely reflected expectations of mutually cooperative outcomes, while Palestinian SGSs exhibited ambivalence and a higher occurrence of confrontational expectations from both parties. Approximately 70% of Israeli SGSs and 40% of Palestinian SGSs were categorized as absolutely stable games, indicating that a meaningful portion of participants implicitly anticipated cooperative and mutually beneficial resolutions. Additionally, Palestinian participants’ perceptions of strategic similarity with Israelis were considerably lower than the perceptions of Israeli participants, pointing to meaningful gaps in the alternatives each side was expecting the other side to choose. The discussion highlights the importance of enhancing subjective perceptions of similarity and shaping parties’ perceived payoff structures as two key pathways to fostering peaceful interactions in diverse social and political conflicts. Keywords: subjective game structures; taxonomy; subjective expected relative similarity; SERS; Israeli–Palestine conflict 1. Introduction International disputes are typically managed and conveyed by central authorities, which define disagreements, shape the public agenda, and drive the use of diplomatic or military actions. However, individuals living in conflict zones may hold perceptions that diverge from those promoted by central authorities. Even within the same group, individuals may experience subjective perceptions that differ from those of other group members or the views expressed by group representatives. Understanding these individually constructed perceptions—particularly those developing within prolonged, ongoing conflicts—can offer valuable insights into the underlying dynamics of the conflict, its potential trajectories, and may suggest innovative approaches for conflict resolution. Previous research has provided participants with a predefined set of possible outcomes, asking them to rate their preferences or utilities. For instance, Plous (1985) collected utility ratings for four Games 2025,16, 3 https://doi.org/10.3390/g16010003 Games 2025,16, 3 2 of 23 proposed outcomes related to the arms race between the US and the Soviet Union during the Cold War. Similarly, Ames et al. (2012) presented American participants with four options describing a US–China conflict, asking them to indicate their own preferences as well as those they believed reflected the average Chinese citizen’s perspective. Halevy et al. (2012) asked participants to rank the outcomes of two-by-two matrices representing negotiation scenarios. While these methods effectively focus participants’ attention on specific outcomes, they do not allow for the elicitation of fully subjective conflict perceptions, which could then be transformed into games and analyzed formally. To this end, we implemented a novel procedure that elicits implicit strategic perspectives from laypeople, requiring no prior knowledge or expertise (Fischer et al.,2024), and translates them into formally defined two-by-two game structures. These resulting games, termed Subjective Game Structures (SGSs), can then be analyzed and classified using various decision-making principles and game-theoretic frameworks. To investigate the Israeli–Palestinian conflict and reveal individual motivations, we sampled participants from Israel, Gaza, and the West Bank. Data collection occurred between 2018 and 2020 as part of an international study applying game theory while aiming to uncover the roots of conflict and war across different global disputes. The results reveal both commonalities and differences in the perspectives of the two populations, suggesting potential interventions that may encourage cooperative solutions. Furthermore, by comparing explicit statements with implicit game structures, we reveal gaps between these perspectives and emphasize the importance of eliciting Subjective Game Structures as a means to better understand and resolve real-life conflicts. Before detailing the experimental procedure, we briefly discuss: (i) the use of game theory as a framework for understanding and analyzing social interactions, conflict, and war; (ii) the calculation of two types of expected values that guide decision-making under uncertainty; and (iii) the classification of games according to two taxonomies: a revised version of the taxonomy proposed by Rapoport and Guyer (1966), and a taxonomy based on the theory of Subjective Expected Relative Similarity (SERS; Fischer,2009,2012;Fischer et al., 2022;Fischer & Savranevski,2023). Readers familiar with some of the topics may wish to skip these sections and proceed directly to the description of the tools and procedures used in this study. Also, readers less interested in applied game-theoretic concepts may prefer to go directly to the sections detailing individual statements and perceptions of similarity. 2. Game-Theoretic Modeling as an Instrument for Understanding and Analyzing Social Interactions Game theory provides useful tools for describing and predicting social and economic interactions. Game matrices describe possible alternatives for two or more parties, and associate them with expected payoffs that may be gained by each of the sides. For example, the well-known Prisoner’s Dilemma game (Flood,1958;Rapoport & Chammah,1965) has been applied to model arms races, the outbreak of war (Axelrod,1984;Brams,1993), and the motivation to pollute and ignore the impact of climate change (Hardin,1968; Ostrom,1998;Fischer et al.,2022). The basic two-by-two PD game provides each party with two alternatives, typically referred to as cooperation and defection. The possible results are described by four different payoffs, termed as follows: Temptation (T), Reward (R), Punishment (P) and Sucker (S), where T > R > P > S (Figure 1a). Examining the PD game reveals several important strategic properties such as the existence of a dominant strategy of defection, and a deficient Nash equilibrium at the intersection of both dominant strategies (see Supplemental Materials—Basic decision criteria). Games 2025,16, 3 3 of 23 Games 2025, 16, x FOR PEER REVIEW 3 of 22 a deficient Nash equilibrium at the intersection of both dominant strategies (see Supplemental Materials—Basic decision criteria). (a) Cooperate Defect Cooperate R, R S, T Defect T, S P, P T > R > P > S (b) Cooperate Defect Cooperate 8, 8 0, 10 Defect 10, 0 1, 1 (c) Cooperate Defect Cooperate R, R S, T Defect T, S P, P T > R > S > P (d) Cooperate Defect Cooperate 8, 8 1, 10 Defect 10, 1 0, 0 Figure 1. Examples of two-by-two game matrices, showing two alternatives for the row and column players and their corresponding payoff values derived from simultaneous choice combinations. Left and right values in each cell indicate the playoffs for the row and column players, respectively. Panel (a) depicts a generic Prisoner’s Dilemma game, defined by the inequalities: T > R > P > S (and in some experiments also 2R > S + T; Flood, 1958; Rapoport & Chammah, 1965). Panel (b) depicts a numeric example of the Prisoner’s Dilemma game. Panel (c) depicts a generic Chicken game, defined by the inequalities: T > R > S > P (Rapoport & Chammah, 1966). Panel (d) depicts a numeric example of the Chicken game. 2.1. The Computation of Expected Values Applying Expected Values (EVs; von Neumann & Morgenstern, 2007) to games, such as the PD game, combines the expected payoffs associated with the choice of each specific alternative with the estimated probabilities of the opponent choosing each alternative. In the case of two-by-two games, the choice of one of the opponent’s alternatives is assigned a probability, p, while the other alternative is assigned the complementary probability of 1 − p. Using the example of the PD game and assigning the probability p to the prospects of the opponent to cooperate allows for the computing the EV for each alternative. The EV associated with own cooperation is given by 𝑅𝑝 + 𝑆(1 − 𝑝), and the EV associated with own defection is given by 𝑇𝑝 + 𝑃(1 − 𝑝) (Figure 1a), letting decision makers choose the alternative with the higher EV. However, it has been shown that when addressing human interactions with an opponent in two-by-two games, assigning a probability to the prospect of the opponent to choose a similar alternative to one’s own chosen alternative, ps (and the complementary probability of 1 − ps to the prospects of the opponent to choose a different alternative), allows for computing EVs that better correspond to actual human choices. This form of EV computation, termed Subjective Expected Relative Similarity (SERS), has been shown to predict actual human behavior in PD, Chicken, and Battle of the Sexes games (Fischer, 2009, 2012; Fischer & Savranevski, 2023, Figure 1), and has been developed into a theoretical strategy that Figure 1. Examples of two-by-two game matrices, showing two alternatives for the row and column players and their corresponding payoff values derived from simultaneous choice combinations. Left and right values in each cell indicate the playoffs for the row and column players, respectively. Panel (a) depicts a generic Prisoner’s Dilemma game, defined by the inequalities: T > R > P > S (and in some experiments also 2R > S + T; Flood, 1958; Rapoport & Chammah, 1965). Panel (b) depicts a numeric example of the Prisoner’s Dilemma game. Panel (c) depicts a generic Chicken game, defined by the inequalities: T > R > S > P (Rapoport & Chammah,1966). Panel (d) depicts a numeric example of the Chicken game. 2.1. The Computation of Expected Values Applying Expected Values (EVs; von Neumann & Morgenstern,2007) to games, such as the PD game, combines the expected payoffs associated with the choice of each specific alternative with the estimated probabilities of the opponent choosing each alternative. In the case of two-by-two games, the choice of one of the opponent’s alternatives is assigned a probability, p, while the other alternative is assigned the complementary probability of 1−p . Using the example of the PD game and assigning the probability pto the prospects of the opponent to cooperate allows for the computing the EV for each alternative. The EV associated with own cooperation is given by Rp +S(1 − p), and the EV associated with own defection is given by Tp +P(1 − p) (Figure 1a), letting decision makers choose the alternative with the higher EV. However, it has been shown that when addressing human interactions with an opponent in two-by-two games, assigning a probability to the prospect of the opponent to choose a similar alternative to one’s own chosen alternative, p s (and the complementary probability of 1 − p s to the prospects of the opponent to choose a different alternative), allows for computing EVs that better correspond to actual human choices. This form of EV computation, termed Subjective Expected Relative Similarity (SERS), has been shown to predict actual human behavior in PD, Chicken, and Battle of the Sexes games (Fischer,2009, 2012;Fischer & Savranevski,2023, Figure 1), and has been developed into a theoretical strategy that maximizes players’ expected outcomes across all two-by-two games (Fischer et al.,2013,2022;Fischer & Avrashi,2024). Games 2025,16, 3 4 of 23 As in the classic EV calculations, SERS predicts that players choose the alternative with the higher EV. For the PD example (where similar choices refer to both mutual cooperation and mutual defection and are assigned the p s probability, and different choices refer to both cases in which one of the players chooses to cooperate and the other chooses to defect, and are therefore assigned the probability of 1 − p s ), a player may, either formally or intuitively, compare the EV of cooperation with the EV of defection and decide to cooperate whenever Rp s +S(1 − p s ) > Pp s +T(1 − p s ). Further separating between perceptions of the opponent,p s , and the game’s payoffs allows for the rewriting of SERS’s decision rule as follows: cooperate whenever p s > (T − S)/(T − S+R − P), otherwise defect. While the left side of the inequality describes the perception of strategic similarity with the opponent, the right side defines a fixed threshold value of the game. Denoting the payoff ratio by p s * generates an abridged decision rule that states the following: cooperate whenever p s >p s *, and defect whenever p s <p s *. Importantly, the p s * index is not only an abridged form of writing, but a critical estimate of the extent of cooperation motivated by each payoff matrix. When p s * is sufficiently low, many opponents may be regarded as being sufficiently similar to motivate the choice of the cooperative alternative; in contrast, a game with a high p s * suggests that most opponents are likely to be considered as insufficiently similar, and therefore motivates the choice of the confrontational alternative. In other words, games with low p s * values may be regarded as more cooperative than games with high p s * values. It has been shown that while strategic similarity is a concealed trait, its extent may be approximated by various observed similarity cues (Chierchia & Coricelli,2015;Fischer,2009, 2012;Toma et al.,2012). In the present study, we ask Israeli and Palestinian participants to assess two similarity estimates: similarity assigned to the other party as a collective group, and similarity assigned to an individual associated with the other party. While group similarity is expected to reflect common stereotypes and perceptions of the outgroup (Banaji,2002), interpersonal similarity is expected to reflect more idiosyncratic, and therefore more heterogeneous, perceptions. Research has shown that participants make more confrontational choices while playing inter-group games than while playing interpersonal games, even when the payoffs in both games are identical (Bornstein & Ben-Yossef,1994). 2.2. Game Classifications While the Prisoner’s Dilemma game used here as an example is a well-known model illustrating strategic properties, it represents just one of many possible games, each with unique strategic characteristics. To identify meaningful strategic similarities and differences between games, one must examine and compare these properties. Accordingly, this study relies on two game classifications: a revised taxonomy of two-by-two games based on Rapoport and Guyer (1966), and a SERS-based taxonomy (Fischer et al.,2024). Rapoport and Guyer’s (1966) taxonomy classifies all strictly ordinal two-by-two games (games in which the payoffs of each player are represented by ordinal ranks—1, 2, 3, 4, each appearing once for each player) into ten categories, characterized by an expected end-state referred to as the natural outcome (which reflects a theoretical expectation of the players to choose an alternative that optimizes their payoffs, see Supplementary Materials—Rapoport and Guyer’s Taxonomy of Two-by-Two Games). Here, we use a revised and abridged taxonomy, which groups all games into five classes (Fischer et al.,2024). This revised taxonomy allows classifying not only strictly ordinal games, but all two-by-two games. The revised taxonomy comprises the following five game categories: (1) Absolutely Stable games, in which both players can jointly obtain their maximal payoffs. Such games are regarded as no-conflict games. In other words, both parties easily and naturally converge on a mutually beneficial solution. (2) Stable/Strongly Stable games, in which one or both players are not Games 2025,16, 3 5 of 23 satisfied with the natural outcome (because they do not obtain their maximal payoff), yet they are not able to influence the outcome of the game by changing, or threatening to change, their choice. In other words, even though there is no mutually beneficial solution where both players obtain their maximal payoff, none of the players is motivated to depart from their initial choice. (3) Non-Stable games, in which one or both players are not satisfied with the natural outcome, yet the player/s are able to influence the game’s outcome by changing or threatening to change their choice. Interactions modeled by these games may be regarded as intractable conflicts, since every action taken by one of the players can be answered with a counteraction of the other player ad infinitum. (4) Prisoners’ Dilemma (PD)-like games 1 , in which neither player is satisfied with the natural outcome, yet they are not motivated to change their choice. Unlike Stable/Strongly Stable games, the natural outcome of PD-like games is a Nash equilibrium but not a Pareto equilibrium 2 . In other words, two rational players are expected to obtain a smaller payoff than the payoff jointly available for both of them in another cell of the matrix. (5) No Natural Outcome games—in this category, one or both players have no preferred choice. Therefore, there is no predicted choice for at least one of the players, and consequently no expected outcome. Figure 2 shows matrix examples for all five classes (see also Supplementary Materials—Rapoport and Guyer’s Taxonomy of Two-by-Two Games). Games 2025, 16, x FOR PEER REVIEW 5 of 22 satisfied with the natural outcome (because they do not obtain their maximal payoff), yet they are not able to influence the outcome of the game by changing, or threatening to change, their choice. In other words, even though there is no mutually beneficial solution where both players obtain their maximal payoff, none of the players is motivated to depart from their initial choice. (3) Non-Stable games, in which one or both players are not satisfied with the natural outcome, yet the player/s are able to influence the game’s outcome by changing or threatening to change their choice. Interactions modeled by these games may be regarded as intractable conflicts, since every action taken by one of the players can be answered with a counteraction of the other player ad infinitum. (4) Prisoners’ Dilemma (PD)- like games1, in which neither player is satisfied with the natural outcome, yet they are not motivated to change their choice. Unlike Stable/Strongly Stable games, the natural outcome of PD-like games is a Nash equilibrium but not a Pareto equilibrium2. In other words, two rational players are expected to obtain a smaller payoff than the payoff jointly available for both of them in another cell of the matrix. (5) No Natural Outcome games—in this category, one or both players have no preferred choice. Therefore, there is no predicted choice for at least one of the players, and consequently no expected outcome. Figure 2 shows matrix examples for all five classes (see also Supplementary Materials—Rapoport and Guyer’s Taxonomy of Two-by-Two Games). Absolutely Stable Stable/Strongly Stable Non-Stable PD-like No Natural Outcome 50, 50 0, 30 30, 0 20, 20 30, 30 50, 20 20, 50 0 , 0 30, 30 20, 50 50, 20 0, 0 20, 20 50, 0 0, 50 30, 30 50, 20 20, 50 20, 50 50, 20 50, 50 0, 20 20, 0 30, 30 20, 30 50, 20 0, 20 30, 0 0, 0 50, 30 30, 50 0, 0 20, 20 30, 0 0, 50 30, 30 50, 0 20, 20 50, 30 20, 50 Figure 2. Example matrices for the five categories of the revised and abridged Rapoport and Guyer taxonomy (Fischer et al., 2024). In all matrices, other than the No Natural Outcome games, the natural outcome is the top left cell. The top Absolutely Stable game is typically referred to as the Stag Hunt game (Skyrms, 2001). The top Non-Stable game is a Chicken game (Rapoport & Chammah, 1966). The bottom Non-Stable game is typically referred to as Battle of the Sexes (Rapoport, 1967). The top PDlike game is a classic PD (Flood, 1958; Rapoport & Chammah, 1965). The top No Natural Outcome game is a Matching Pennies game (Budescu & Rapoport, 1994). While the revised Rapoport and Guyer taxonomy helps gain theoretical insights, the success of SERS in predicting behavioral choices (Fischer, 2009, 2012; Fischer et al., 2022; Fischer & Savranevski, 2023) suggests also using it as a basis for a taxonomy of games. As mentioned above, SERS shifts the focus from the payoff structure per se to the interaction between the game’s payoff structure and the players’ perceptions of their opponent; specifically, the prospects of both players making similar choices. Therefore, the SERS-based taxonomy distinguishes between two basic types of payoff sets. Games in which the SERSbased expected choices vary under different perceptions of strategic similarity with the opponent are referred to as similarity-sensitive games. Games in which the SERS-based expected choices do not vary under different perceptions of strategic similarity with the opponent are referred to as non-similarity-sensitive games. Clearly, some games can be similarity-sensitive for one of the players and non-similarity-sensitive for the other. Therefore, the SERS-based Figure 2. Example matrices for the five categories of the revised and abridged Rapoport and Guyer taxonomy (Fischer et al.,2024). In all matrices, other than the No Natural Outcome games, the natural outcome is the top left cell. The top Absolutely Stable game is typically referred to as the Stag Hunt game (Skyrms,2001). The top Non-Stable game is a Chicken game (Rapoport & Chammah,1966). The bottom Non-Stable game is typically referred to as Battle of the Sexes (Rapoport,1967). The top PD-like game is a classic PD (Flood,1958;Rapoport & Chammah,1965). The top No Natural Outcome game is a Matching Pennies game (Budescu & Rapoport,1994). While the revised Rapoport and Guyer taxonomy helps gain theoretical insights, the success of SERS in predicting behavioral choices (Fischer,2009,2012;Fischer et al., 2022;Fischer & Savranevski,2023) suggests also using it as a basis for a taxonomy of games. As mentioned above, SERS shifts the focus from the payoff structure per se to the interaction between the game’s payoff structure and the players’ perceptions of their opponent; specifically, the prospects of both players making similar choices. Therefore, the SERS-based taxonomy distinguishes between two basic types of payoff sets. Games in which the SERS-based expected choices vary under different perceptions of strategic similarity with the opponent are referred to as similarity-sensitive games. Games in which the SERS-based expected choices do not vary under different perceptions of strategic similarity with the opponent are referred to as non-similarity-sensitive games. Clearly, some games can be similarity-sensitive for one of the players and non-similarity-sensitive for the other. Games 2025,16, 3 6 of 23 Therefore, the SERS-based taxonomy differentiates between two-player similarity-sensitive games,one-player similarity-sensitive games, and two-player non-similarity-sensitive games. The importance of this classification for analysis of games derives from its capacity to determine which games are susceptible to social and behavioral interventions, allowing for the influencing of the parties’ choices. That is, influencing perceptions of strategic similarity between the interacting parties (in the present study—Israelis and Palestinians) has the potential to alter the expected outcome of similarity-sensitive games without changing the game’s actual payoffs. See Supplementary Materials—SERS-Based Taxonomy of Games for a detailed description of the SERS-based classification. Figure 3shows examples of games associated with each of the four classes of the SERS-based classification. Games 2025, 16, x FOR PEER REVIEW 6 of 22 taxonomy differentiates between two-player similarity-sensitive games, one-player similaritysensitive games, and two-player non-similarity-sensitive games. The importance of this classification for analysis of games derives from its capacity to determine which games are susceptible to social and behavioral interventions, allowing for the influencing of the parties’ choices. That is, influencing perceptions of strategic similarity between the interacting parties (in the present study—Israelis and Palestinians) has the potential to alter the expected outcome of similarity-sensitive games without changing the game’s actual payoffs. See Supplementary Materials—SERS-Based Taxonomy of Games for a detailed description of the SERS-based classification. Figure 3 shows examples of games associated with each of the four classes of the SERS-based classification. (a) A B A 20, 20 10, 10 B 15, 15 5, 5 (b) A B A 15, 20 20, 15 B 10, 10 5, 5 (c) A B A 15, 15 5, 20 B 20, 5 10, 10 (d) A B A 15, 15 20, 10 B 10, 20 5, 5 Figure 3. Examples of the four classes of the SERS-based classification. (a) depicts a row player-only similarity-sensitive game. When assuming both players are completely similar, and thus both are expected to choose the same alternatives (either AA or BB), the row player prefers 20 over 5, hence choosing the top row. However, when assuming both players are completely dissimilar, and thus both are expected to choose opposite alternatives (either AB or BA), the row player prefers 15 over 10, and hence chooses the bottom row. Consequently, similarity perceptions change the preferred alternative of the row player. In contrast, the column player prefers the left column under both assumptions of complete similarity and complete dissimilarity (since 20 is preferred over 5 for complete similarity, and 15 is preferred over 10 for complete dissimilarity). Hence, similarity perceptions do not change the preferred alternative of the column player. (b) depicts a column player-only similarity-sensitive game, as the row player prefers the upper row under both the assumptions of complete similarity and complete dissimilarity, while the column player prefers the left column under the assumption of complete similarity, and the right column under the assumption of complete dissimilarity. (c) (also termed the Prisoner’s Dilemma game) depicts a two-player similarity-sensitive game. Both players prefer alternative A under the assumption of complete similarity and alternative B under the assumption of complete dissimilarity. (d) depicts a two-player non-similarity-sensitive. Both players have the same preferences under the assumption of complete similarity and the assumption of complete dissimilarity. Note that complete similarity and dissimilarity are two extreme cases helping to easily determine the class of the game. Actual similarity perceptions may obtain any value between, and including, these two extremes. 3. Methods Participants: One hundred and forty-one Israeli Jewish individuals (50% women, Mage = 24.6, SD = 4.86), recruited at an Israeli university, participated in the study in award of academic credit points or the equivalent of USD 20. Eighty-six Palestinian individuals (47% women, Mage = 36.7, SD = 11.97) from the Gaza Strip and the West Bank who visited Israel for personal reasons participated in the study in award of the equivalent of USD 20. Data were collected between March 2019 and February 2020. Figure 3. Examples of the four classes of the SERS-based classification. (a) depicts a row playeronly similarity-sensitive game. When assuming both players are completely similar, and thus both are expected to choose the same alternatives (either AA or BB), the row player prefers 20 over 5, hence choosing the top row. However, when assuming both players are completely dissimilar, and thus both are expected to choose opposite alternatives (either AB or BA), the row player prefers 15 over 10, and hence chooses the bottom row. Consequently, similarity perceptions change the preferred alternative of the row player. In contrast, the column player prefers the left column under both assumptions of complete similarity and complete dissimilarity (since 20 is preferred over 5 for complete similarity, and 15 is preferred over 10 for complete dissimilarity). Hence, similarity perceptions do not change the preferred alternative of the column player. (b) depicts a column playeronly similarity-sensitive game, as the row player prefers the upper row under both the assumptions of complete similarity and complete dissimilarity, while the column player prefers the left column under the assumption of complete similarity, and the right column under the assumption of complete dissimilarity. (c) (also termed the Prisoner’s Dilemma game) depicts a two-player similarity-sensitive game. Both players prefer alternative A under the assumption of complete similarity and alternative B under the assumption of complete dissimilarity. (d) depicts a two-player non-similarity-sensitive. Both players have the same preferences under the assumption of complete similarity and the assumption of complete dissimilarity. Note that complete similarity and dissimilarity are two extreme cases helping to easily determine the class of the game. Actual similarity perceptions may obtain any value between, and including, these two extremes. 3. Methods Participants: One hundred and forty-one Israeli Jewish individuals (50% women, M age = 24.6, SD = 4.86), recruited at an Israeli university, participated in the study in award of academic credit points or the equivalent of USD 20. Eighty-six Palestinian individuals (47% women, M age = 36.7, SD = 11.97) from the Gaza Strip and the West Bank who visited Israel for personal reasons participated in the study in award of the equivalent of USD 20. Data were collected between March 2019 and February 2020. Games 2025,16, 3 7 of 23 3.1. Tools Symbolic Quantification Tools (SQT): visual aids that allow the participants to express quantities, using intuitive pictorial representations that do not necessitate formal numerical skills (Fischer et al.,2024). The two SQTs consist of a series of illustrations, 24 showing various depictions of a tree, ranging from a worst condition (a tree with no leaves at all), to a best condition (a tree with a rich foliage); and 41 showing depictions of a jar filled with various portions of a gold-colored substance. The tree and jar SQTs were selected from among six SQTs that had been validated in a preceding study, which associated each image with a specific numerical value. They were selected for inclusion in the present study due to their relatively high convergent validity and the consistently perceived intervals between successive illustrations. See Figure 4for examples of the SQT illustrations. Games 2025, 16, x FOR PEER REVIEW 7 of 22 3.1. Tools Symbolic Quantification Tools (SQT): visual aids that allow the participants to express quantities, using intuitive pictorial representations that do not necessitate formal numerical skills (Fischer et al., 2024). The two SQTs consist of a series of illustrations, 24 showing various depictions of a tree, ranging from a worst condition (a tree with no leaves at all), to a best condition (a tree with a rich foliage); and 41 showing depictions of a jar filled with various portions of a gold-colored substance. The tree and jar SQTs were selected from among six SQTs that had been validated in a preceding study, which associated each image with a specific numerical value. They were selected for inclusion in the present study due to their relatively high convergent validity and the consistently perceived intervals between successive illustrations. See Figure 4 for examples of the SQT illustrations. 0 11.7 15.5 33.6 43.1 66.3 89.1 100 0 12.5 27.5 40 55 67.5 85 100 Figure 4. A representative subset of the tree and jar illustrations used as SQTs and their associated values. The top values correspond to the tree SQT images and the bottom values to the jar SQT images. 3.2. Procedure Participants were welcomed by a research assistant who spoke their native language (Hebrew or Arabic) and asked if they could spare approximately one hour of their time. All participants were shown and read a description of their rights as participants, assured of full anonymity during and after the study, and of their right to discontinue their participation without any consequences. They were then seated in front of a computer with dedicated software that presented the questions and SQTs, and collected participants’ responses. The experiment started by mentioning the ongoing Israeli–Palestinian conflict and asking the participants to suggest two different/opposing and realistic courses of action, first for their own side (either Israeli or Palestinian), and then for the other side (either Palestinian or Israeli). To assure that the meaning of opposing alternatives was linguistically and conceptually comprehended by Arabic speaking participants, the Arabic version also included a short example describing two cellphone companies considering how to take over a larger market share, by examining opposing alternatives (i.e., selling at a fair price or selling at a lower price than the competitor; praising their own product or defaming the competitor). Notice that the instructions did not explicitly mention the terms ‘cooperation’ and ‘confrontation’, thus avoiding induction of a specific framing that might not coincide with participants’ genuine attitudes and reasoning. After the participants defined their own alternatives, the software presented the four combinations generated by each participant’s provided alternatives, and asked participants to list three possible outcomes, experiences, or emotions that are likely to be experienced by the participant as a member of his or her group (separately for each intersection of alternatives). The same set of four questions was repeated while asking the participant to describe the outcomes, experiences, and emotions the other party is likely to experience. Next, each of the Figure 4. A representative subset of the tree and jar illustrations used as SQTs and their associated values. The top values correspond to the tree SQT images and the bottom values to the jar SQT images. 3.2. Procedure Participants were welcomed by a research assistant who spoke their native language (Hebrew or Arabic) and asked if they could spare approximately one hour of their time. All participants were shown and read a description of their rights as participants, assured of full anonymity during and after the study, and of their right to discontinue their participation without any consequences. They were then seated in front of a computer with dedicated software that presented the questions and SQTs, and collected participants’ responses. The experiment started by mentioning the ongoing Israeli–Palestinian conflict and asking the participants to suggest two different/opposing and realistic courses of action, first for their own side (either Israeli or Palestinian), and then for the other side (either Palestinian or Israeli). To assure that the meaning of opposing alternatives was linguistically and conceptually comprehended by Arabic speaking participants, the Arabic version also included a short example describing two cellphone companies considering how to take over a larger market share, by examining opposing alternatives (i.e., selling at a fair price or selling at a lower price than the competitor; praising their own product or defaming the competitor). Notice that the instructions did not explicitly mention the terms ‘cooperation’ and ‘confrontation’, thus avoiding induction of a specific framing that might not coincide with participants’ genuine attitudes and reasoning. After the participants defined their own alternatives, the software presented the four combinations generated by each participant’s provided alternatives, and asked participants to list three possible outcomes,experiences,or emotions that are likely to be experienced by the participant as a member of his or her group (separately for each intersection of alternatives). The same set of four questions was repeated while asking the participant to describe the outcomes, experiences, and emotions the other party is likely to experience. Next, each of the two SQTs was presented and explained, explicitly providing examples Games 2025,16, 3 8 of 23 of the best and worst outcomes and their respective images (Figure 4). After the SQT introduction, each participant was asked to separately consider the four Israeli–Palestinian alternative intersections (as proposed by the participant), each presented alongside the three consequences (also provided by the participant), and to choose the SQT image that best represents the desirability of the situation, once for the tree SQT and once for the jar SQT. The process was then repeated, asking the participant to assess the desirability of the four alternative intersections and their respective three consequences while considering the other party (either the Israeli or the Palestinian party). The next couple of questions addressed participants’ forecasts, asking them to identify which of the two alternatives they suggested earlier for their own party is more likely to be enacted, and which of the two alternatives they suggested for the other party is more likely to be enacted by the other party. Next, two questions addressed participants’ subjective perceptions of similarity with the other party. First, participants were asked to think of a person who best represents the other side and indicate, using a slider, how similar this person is to themselves (i.e., interpersonal similarity). Second, participants indicated how similar the other group is to their own group (i.e., group similarity), again using a slider. Finally, basic demographic information was collected. Palestinian participants were debriefed and received their participation fee. Israeli participants were invited to a second session, taking place about a week later, which allowed further testing the reliability of the SQT elicitation method 3 (Fischer et al.,2024). The data were collected between 2018 and 2020. See Figure 5for an illustration of the procedure. Games 2025, 16, x FOR PEER REVIEW 8 of 22 two SQTs was presented and explained, explicitly providing examples of the best and worst outcomes and their respective images (Figure 4). After the SQT introduction, each participant was asked to separately consider the four Israeli–Palestinian alternative intersections (as proposed by the participant), each presented alongside the three consequences (also provided by the participant), and to choose the SQT image that best represents the desirability of the situation, once for the tree SQT and once for the jar SQT. The process was then repeated, asking the participant to assess the desirability of the four alternative intersections and their respective three consequences while considering the other party (either the Israeli or the Palestinian party). The next couple of questions addressed participants’ forecasts, asking them to identify which of the two alternatives they suggested earlier for their own party is more likely to be enacted, and which of the two alternatives they suggested for the other party is more likely to be enacted by the other party. Next, two questions addressed participants’ subjective perceptions of similarity with the other party. First, participants were asked to think of a person who best represents the other side and indicate, using a slider, how similar this person is to themselves (i.e., interpersonal similarity). Second, participants indicated how similar the other group is to their own group (i.e., group similarity), again using a slider. Finally, basic demographic information was collected. Palestinian participants were debriefed and received their participation fee. Israeli participants were invited to a second session, taking place about a week later, which allowed further testing the reliability of the SQT elicitation method 3 (Fischer et al., 2024). The data were collected between 2018 and 2020. See Figure 5 for an illustration of the procedure. Figure 5. Illustration of the procedure allowing the eliciting of Subjective Game Structures (SGSs) while using the tree SQT. Figure 5. Illustration of the procedure allowing the eliciting of Subjective Game Structures (SGSs) while using the tree SQT. Games 2025,16, 3 15 of 23 Games 2025, 16, x FOR PEER REVIEW 14 of 22 the Israeli SGSs comprise 72% absolutely stable games and only 22% non-stable games, Palestinian SGSs comprise 41% absolutely stable games and a rather equal proportion, 35%, of non-stable games. Out of the Israeli absolutely stable games, 95% reflect a mutually cooperative solution that endows both parties with their maximal expected payoff. This property also holds for 93% of the Palestinian absolutely stable games. Therefore, almost all elicited absolutely stable games express the expectation for a mutually cooperative solution. However, the gap in the proportions of absolutely stable games between the parties (72% vs. 41%) reveals meaningful differences in the perception of the Israeli-Palestinian conflict. Noticeably, none of the parties tend to perceive the interaction as a PD-like game. Israeli sample Palestinian sample Figure 9. Distributions of Israeli and Palestinian SGSs according to the revised Rapoport and Guyer taxonomy of games. v. Distribution of game types according to the SERS-based taxonomy Further examining the SERS-based taxonomy of all games, Figure 10 shows that about a third of all games in both samples are two-player similarity-sensitive games. However, Israeli participants perceived almost three times as many Israeli one-sided similarity-sensitive games (26%) compared to Palestinian participants’ perceived Israeli one-sided similaritysensitive games (9%). Also, Israeli participants perceived fewer Palestinian one-sided similarity-sensitive games (27%) compared to Palestinian participants’ perceived Palestinian one-sided similarity-sensitive games (43%). Overall, the distributions of SERS-based game types reveal a significant difference between the perceptions of both parties (χ 2 = 19.47, p < 0.001). Israeli sample Palestinian sample Figure 10. Distributions of Israeli and Palestinian SGSs according to the SERS-based taxonomy of games. Figure 9. Distributions of Israeli and Palestinian SGSs according to the revised Rapoport and Guyer taxonomy of games. v. Distribution of game types according to the SERS-based taxonomy Further examining the SERS-based taxonomy of all games, Figure 10 shows that about a third of all games in both samples are two-player similarity-sensitive games. However, Israeli participants perceived almost three times as many Israeli one-sided similaritysensitive games (26%) compared to Palestinian participants’ perceived Israeli one-sided similarity-sensitive games (9%). Also, Israeli participants perceived fewer Palestinian one-sided similarity-sensitive games (27%) compared to Palestinian participants’ perceived Palestinian one-sided similarity-sensitive games (43%). Overall, the distributions of SERSbased game types reveal a significant difference between the perceptions of both parties (χ2= 19.47, p< 0.001). Games 2025, 16, x FOR PEER REVIEW 14 of 22 the Israeli SGSs comprise 72% absolutely stable games and only 22% non-stable games, Palestinian SGSs comprise 41% absolutely stable games and a rather equal proportion, 35%, of non-stable games. Out of the Israeli absolutely stable games, 95% reflect a mutually cooperative solution that endows both parties with their maximal expected payoff. This property also holds for 93% of the Palestinian absolutely stable games. Therefore, almost all elicited absolutely stable games express the expectation for a mutually cooperative solution. However, the gap in the proportions of absolutely stable games between the parties (72% vs. 41%) reveals meaningful differences in the perception of the Israeli-Palestinian conflict. Noticeably, none of the parties tend to perceive the interaction as a PD-like game. Israeli sample Palestinian sample Figure 9. Distributions of Israeli and Palestinian SGSs according to the revised Rapoport and Guyer taxonomy of games. v. Distribution of game types according to the SERS-based taxonomy Further examining the SERS-based taxonomy of all games, Figure 10 shows that about a third of all games in both samples are two-player similarity-sensitive games. However, Israeli participants perceived almost three times as many Israeli one-sided similarity-sensitive games (26%) compared to Palestinian participants’ perceived Israeli one-sided similaritysensitive games (9%). Also, Israeli participants perceived fewer Palestinian one-sided similarity-sensitive games (27%) compared to Palestinian participants’ perceived Palestinian one-sided similarity-sensitive games (43%). Overall, the distributions of SERS-based game types reveal a significant difference between the perceptions of both parties (χ 2 = 19.47, p < 0.001). Israeli sample Palestinian sample Figure 10. Distributions of Israeli and Palestinian SGSs according to the SERS-based taxonomy of games. Figure 10. Distributions of Israeli and Palestinian SGSs according to the SERS-based taxonomy of games. Merging two-player similarity-sensitive games with own-side similarity-sensitive games, separately for each party, shows that 64% of the Israeli SGSs are similarity-sensitive for the Israeli side, and that 76% of the Palestinian SGSs are similarity-sensitive for the Palestinian side. These relatively high percentages point to the importance of similarity perceptions, suggesting that influencing the parties’ similarity perceptions of each other may play a key role in resolving the conflict. vi. Perceived inter-group and interpersonal similarity distributions As mentioned above, participants provided two types of similarity perceptions, one that addresses the similarity between the two groups, and one that addresses the similarity between the participant and a person the participant considers to be representative of Games 2025,16, 3 16 of 23 the other group. Figure 11 shows that both inter-group and interpersonal similarities reported by the Israeli participants give rise to a rather homogeneous distribution, where all similarity levels (apart from extremely high values) are equally represented. In contrast, the distributions of Palestinian inter-group and interpersonal similarities are characterized by a large proportion, about 40%, of extremely low similarity perceptions (ranging from 0 to 10, out of 100); while the other 60% of reported similarity perceptions exhibit a rather homogeneous distribution across all other similarity levels. For both groups, inter-group and interpersonal similarity assessments reveal identical distributions (Figure 11). The average Israeli participants’ inter-group and interpersonal similarity estimates are 40.0 (SD = 25.04) and 39.4 (SD = 26.08), respectively; and the correlation between both measures is r = 0.55 (t (140) =7.77, p< 0.001). The Palestinian participants’ inter-group and interpersonal similarity estimates are 31.3 (SD = 30.60) and 36.7 (SD = 35.36), respectively; and the correlation between both measures is r = 0.40 (t (85) = 3.95, p< 0.001). Comparing the average reported similarity estimates of Israeli and Palestinian participants shows a significant difference for the inter-group similarity (t (153) = 2.22, p= 0.028; Cohen’s d = 0.32), and no significant difference for the interpersonal similarity measure (t (141) = 0.60, n.s.; Cohen’s d = 0.09). Focusing on the inter-group conflict, we continue the analyses by using the inter-group similarity measure in SERS-based EV calculations. Games 2025, 16, x FOR PEER REVIEW 16 of 22 Israeli inter-group similarity Palestinian inter-group similarity Israeli interpersonal similarity Palestinian interpersonal similarity Figure 11. Reported similarity distributions for all participants. N Israeli = 141; N Palestinian = 86. Figure 12. Cooperative and confrontational predicted choices, derived from SERS-based EVs. viii. Averaged SGSs grouped by perceived similarity quartiles According to SERS, (i) people with high similarity perceptions of the opponent are more likely to cooperate than people with low similarity perceptions, and (ii) people whose SGSs have low similarity thresholds are more likely to cooperate than people whose SGSs have high similarity thresholds. This raises the question of whether there exists a behavioral link between both perceptions, where the same participants that report high similarity levels with the opponent also provide SGSs with low similarity thresholds, and vice versa. To answer this question, we split the participants’ samples into quartiles, according to the reported similarity perceptions. We then calculate the average SGSs separately for each similarity quartile, and describe each averaged SGS in terms of the two taxonomies and in terms of the game’s (own and opponent) similarity thresholds. Note that quartiles are calculated across all Figure 11. Reported similarity distributions for all participants. NIsraeli = 141; NPalestinian = 86. vii. Predicted choices according to SERS Examining the SGSs and the inter-group similarity level reported by each participant allows for the computing of two SERS-based EVs (i.e., a cooperative and a confrontational EV) and choosing the alternative with the higher EV, first for the participant’s own party, and then for the opponent. Note that the perceived strategic similarity is regarded here as a symmetric property. Therefore, the reported p s of each participant is used to calculate one’s own SERS-based expected value as well as the opponent’s SERS-based expected value. Games 2025,16, 3 17 of 23 Figure 12 shows that the proportion of Israeli SERS-based cooperative choices for their own (Israeli) side is much higher (66%) than the Palestinian SERS-based cooperative choices for their own (Palestinian) side (49%; χ2 = 8.00, p< 0.01). Moreover, the proportion of Israeli SERS-based cooperative choices for their own (Israeli) side is much higher (66%) than the Palestinian SERS-based cooperative choices for the other (Israeli) side (42%), resulting in a significantly different distribution of cooperative and confrontational expected choices ( χ2 = 9.86, p< 0.01). In other words, according to SERS, Israelis are predicted to choose more cooperative choices than the Palestinians expect them to choose. In contrast, the proportion of Palestinian SERS-based cooperative choices for their own (Palestinian) side is lower (49%) than the proportion of Israeli SERS-based cooperative choices for the other (Palestinian) side (58%). However, the distributions of cooperative and confrontational expected choices for the Palestinian side are not significantly different when comparing the Israeli and Palestinian samples (χ2= 2.87, p= 0.09). Games 2025, 16, x FOR PEER REVIEW 16 of 22 Israeli inter-group similarity Palestinian inter-group similarity Israeli interpersonal similarity Palestinian interpersonal similarity Figure 11. Reported similarity distributions for all participants. N Israeli = 141; N Palestinian = 86. Figure 12. Cooperative and confrontational predicted choices, derived from SERS-based EVs. viii. Averaged SGSs grouped by perceived similarity quartiles According to SERS, (i) people with high similarity perceptions of the opponent are more likely to cooperate than people with low similarity perceptions, and (ii) people whose SGSs have low similarity thresholds are more likely to cooperate than people whose SGSs have high similarity thresholds. This raises the question of whether there exists a behavioral link between both perceptions, where the same participants that report high similarity levels with the opponent also provide SGSs with low similarity thresholds, and vice versa. To answer this question, we split the participants’ samples into quartiles, according to the reported similarity perceptions. We then calculate the average SGSs separately for each similarity quartile, and describe each averaged SGS in terms of the two taxonomies and in terms of the game’s (own and opponent) similarity thresholds. Note that quartiles are calculated across all Figure 12. Cooperative and confrontational predicted choices, derived from SERS-based EVs. viii. Averaged SGSs grouped by perceived similarity quartiles According to SERS, (i) people with high similarity perceptions of the opponent are more likely to cooperate than people with low similarity perceptions, and (ii) people whose SGSs have low similarity thresholds are more likely to cooperate than people whose SGSs have high similarity thresholds. This raises the question of whether there exists a behavioral link between both perceptions, where the same participants that report high similarity levels with the opponent also provide SGSs with low similarity thresholds, and vice versa. To answer this question, we split the participants’ samples into quartiles, according to the reported similarity perceptions. We then calculate the average SGSs separately for each similarity quartile, and describe each averaged SGS in terms of the two taxonomies and in terms of the game’s (own and opponent) similarity thresholds. Note that quartiles are calculated across all participants, yet average SGSs comprise only alignable SGSs. Also note that 31% of the Palestinian sample reported a similarity level of 0; therefore, the lower similarity quartile comprises more SGSs compared to the other quartiles. Figure 13 shows that all averaged matrices of both the Israeli and Palestinian samples, apart from the Palestinian lowest similarity quartile (Q1), are absolutely stable games according to the revised Rapoport and Guyer’s taxonomy. For these games, the maximal payoffs are earned when both parties choose to cooperate. Only the Palestinian Q1 averaged SGS is a non-stable game, which is likely to motivate an intractable conflict. Moreover, all averaged SGSs of both parties for all quartiles are two-player similarity-sensitive games, meaning that both parties’ strategic similarity perceptions are critical for determining the parties’ expected choices. Games 2025,16, 3 18 of 23 Games 2025, 16, x FOR PEER REVIEW 17 of 22 participants, yet average SGSs comprise only alignable SGSs. Also note that 31% of the Palestinian sample reported a similarity level of 0; therefore, the lower similarity quartile comprises more SGSs compared to the other quartiles. Figure 13 shows that all averaged matrices of both the Israeli and Palestinian samples, apart from the Palestinian lowest similarity quartile (Q1), are absolutely stable games according to the revised Rapoport and Guyer’s taxonomy. For these games, the maximal payoffs are earned when both parties choose to cooperate. Only the Palestinian Q1 averaged SGS is a non-stable game, which is likely to motivate an intractable conflict. Moreover, all averaged SGSs of both parties for all quartiles are two-player similarity-sensitive games, meaning that both parties’ strategic similarity perceptions are critical for determining the parties’ expected choices. a. Israeli Q1 mean matrix (11 participants, 22 matrices) Pal. C Pal. D Isr. C 78.0, 73.3 15.2, 52.4 Isr. D 53.8, 16.9 26.2, 39.0 Reported similarity range: 0–0.20 Israel ps* = 0.43 Palestine ps* = 0.51 b. Israeli Q2 mean matrix (22 participants, 44 matrices) Pal. C Pal. D Isr. C 87.5, 76.1 25.2, 42.1 Isr. D 43.6, 21.4 21.8, 21.9 Reported similarity range: 0.21–0.36 Israel ps* = 0.22 Palestine ps* = 0.28 c. Israeli Q3 mean matrix (14 participants, 28 matrices) Pal. C Pal. D Isr. C 68.6, 61.9 23, 39.3 Isr. D 33.5, 23.3 14, 20.6 Reported similarity range: 0.37–0.59 Israel ps* = 0.16 Palestine ps* = 0.28 d. Israeli Q4 mean matrix (20 participants, 40 matrices) Pal. C Pal. D Isr. C 85.6, 73.4 29.7, 48.5 Isr. D 32.6, 21.8 17.2, 25.0 Reported similarity range: 0.60–0.94 Israel ps* = 0.04 Palestine ps* = 0.36 e. Palestinian Q1 mean matrix (24 participants, 48 matrices) Pal. C Pal. D Isr. C 60.7, 88.3 32.9, 62.4 Isr. D 68.6, 9.1 47.8, 5.2 Reported similarity range: 0.00–0.00 Israel ps* = 0.73 Palestine ps* = 0.39 f. Palestinian Q2 mean matrix (15 participants, 30 matrices) Pal. C Pal. D Isr. C 72.3, 74.7 38, 33.6 Isr. D 48.2, 19.4 31.2, 15.0 Reported similarity range: 0.01–0.22 Israel ps* = 0.20 Palestine ps* = 0.19 Games 2025, 16, x FOR PEER REVIEW 18 of 22 g. Palestinian Q3 mean matrix (20 participants, 40 matrices) Pal. C Pal. D Isr. C 59.4, 81.6 36.3, 60.8 Isr. D 49.0, 18.4 39.6, 11.8 Reported similarity range: 0.23–0.54 Israel ps* = 0.39 Palestine ps* = 0.38 h. Palestinian Q4 mean matrix (16 participants, 32 matrices) Pal. C Pal. D Isr. C 73.5, 75.6 23.9, 37.0 Isr. D 48.5, 9.1 27.9, 7.3 Reported similarity range: 0.55–1.00 Israel ps* = 0.35 Palestine ps* = 0.29 Figure 13. Average SGSs grouped by reported inter-group similarity quartiles for Israeli and Palestinians participants. As shown in Figure 13, the similarity thresholds of the Israeli side strictly decrease across the four quartiles. The similarity threshold of those who perceive low similarity with the Palestinians (Q1) is ps* = 0.43. Moving through the matrices of Q2 to Q4 reveals a gradual drop to ps* = 0.04. Comparing these similarity thresholds with the corresponding similarity perceptions in each quartile suggests that ps > ps* for all averaged SGSs apart from Q1 (where 0.2 < 0.43). Therefore, most Israeli participants are expected to choose the cooperative alternative. Examining the Palestinian participants’ sample, reveals an opposite pattern, where all averaged SGSs except Q4 are characterized by ps < ps*, suggesting the choice of defection for most Palestinian participants. 5. Discussion The present study examined the Israeli–Palestinian conflict (as perceived about three years prior to the outbreak of the 2023 Gaza war). Instead of relying on scholarly or political views, we applied a novel paradigm that allowed for eliciting implicit conflict perceptions, expressed as Subjective Game Structures (SGSs) from Israeli and Palestinian laymen, requiring no formal quantitative background for the participants. We then applied several analytical tools, allowing for underlying subjective perceptions of the conflict to be considered. Moreover, we tried to answer the questions of whether Israelis and Palestinians perceive the conflict in a similar manner; whether they have identical assessments of the associated payoffs; and most critically—what can be done to reduce future hostilities and attain peaceful solutions? The results show that Israeli and Palestinian participants differed in their perceptions of the conflict. Israeli perceptions reflected rather cooperative views and the expectation for mutually cooperative resolutions, whereas Palestinian perceptions reflected ambivalent attitudes, with a higher proportion of confrontational expectations from both Israelis and Palestinians. Initial examination of participants’ self-reports showed that most Israeli participants reported cooperative predictions for both sides (80% for Israelis and 75% for Palestinians), while Palestinian cooperative and confrontational expectations were more evenly divided (48% cooperation for Israelis and 55% for Palestinians). Further analyzing participants’ elicited SGSs showed that the averaged Israeli and Palestinian SGSs are both classified as absolutely stable games, which in theory should motivate fully cooperative interactions. Nevertheless, examining the same payoffs from the perspective of SERS (Fischer, 2009, 2012; Fischer et al., 2022; Fischer & Savranevski, 2023) reveals meaningful and critical differences between the parties’ average matrices. Although both average matrices are two-player similarity-sensitive games, the Israeli matrix has very Figure 13. Average SGSs grouped by reported inter-group similarity quartiles for Israeli and Palestinians participants. Games 2025,16, 3 19 of 23 As shown in Figure 13, the similarity thresholds of the Israeli side strictly decrease across the four quartiles. The similarity threshold of those who perceive low similarity with the Palestinians (Q1) is p s * = 0.43. Moving through the matrices of Q2 to Q4 reveals a gradual drop to p s * = 0.04. Comparing these similarity thresholds with the corresponding similarity perceptions in each quartile suggests that p s >p s * for all averaged SGSs apart from Q1 (where 0.2 < 0.43). Therefore, most Israeli participants are expected to choose the cooperative alternative. Examining the Palestinian participants’ sample, reveals an opposite pattern, where all averaged SGSs except Q4 are characterized by p s <p s *, suggesting the choice of defection for most Palestinian participants. 5. Discussion The present study examined the Israeli–Palestinian conflict (as perceived about three years prior to the outbreak of the 2023 Gaza war). Instead of relying on scholarly or political views, we applied a novel paradigm that allowed for eliciting implicit conflict perceptions, expressed as Subjective Game Structures (SGSs) from Israeli and Palestinian laymen, requiring no formal quantitative background for the participants. We then applied several analytical tools, allowing for underlying subjective perceptions of the conflict to be considered. Moreover, we tried to answer the questions of whether Israelis and Palestinians perceive the conflict in a similar manner; whether they have identical assessments of the associated payoffs; and most critically—what can be done to reduce future hostilities and attain peaceful solutions? The results show that Israeli and Palestinian participants differed in their perceptions of the conflict. Israeli perceptions reflected rather cooperative views and the expectation for mutually cooperative resolutions, whereas Palestinian perceptions reflected ambivalent attitudes, with a higher proportion of confrontational expectations from both Israelis and Palestinians. Initial examination of participants’ self-reports showed that most Israeli participants reported cooperative predictions for both sides (80% for Israelis and 75% for Palestinians), while Palestinian cooperative and confrontational expectations were more evenly divided (48% cooperation for Israelis and 55% for Palestinians). Further analyzing participants’ elicited SGSs showed that the averaged Israeli and Palestinian SGSs are both classified as absolutely stable games, which in theory should motivate fully cooperative interactions. Nevertheless, examining the same payoffs from the perspective of SERS (Fischer,2009,2012;Fischer et al.,2022;Fischer & Savranevski, 2023) reveals meaningful and critical differences between the parties’ average matrices. Although both average matrices are two-player similarity-sensitive games, the Israeli matrix has very low similarity thresholds of p s * = 0.20 for the Israeli side and p s * = 0.34 for the Palestinian side, whereas the similarity thresholds of the Palestinian matrix are p s * = 0.45 for the Israeli side, and p s * = 0.34 for the Palestinian side. Interpreting these values requires comparing them with the parties’ similarity perceptions of each other. The average reported inter-group similarities for the Israeli and Palestinian parties are ps = 0.40 and ps = 0.31, respectively. Therefore, despite both matrices being rather identical, one may expect a cooperative choice made by the Israeli side (since 0.40 > 0.20) yet a rather inconclusive tendency for the Palestinian side (since 0.31 ≈0.34). Examining the SGSs of individuals with different self-reported conflict expectations shows that individuals from both samples who expect Israeli cooperation also provide matrices with very low similarity thresholds for the Israeli party; while those who expected Israeli confrontation provided matrices with high similarity thresholds for the Israeli party. A similar pattern, though somewhat less extreme, is also observed when considering the self-reported conflict expectations and the similarity thresholds of Palestinian participants. In other words, individuals who reported expecting cooperation have a more cooperative Games 2025,16, 3 20 of 23 perception of the conflict structure itself (i.e., requiring only a low level of perceived similarity with the opponent in order to cooperate according to SERS). In parallel, those who report confrontational expectations have more confrontational perceptions of the conflict structure (i.e., requiring a high level of perceived similarity with the opponent in order to cooperate). This pattern is clearly demonstrated by Palestinian participants who predicted Israeli confrontation and provided SGSs with extremely high similarity thresholds for the Israeli party (p s * = 0.93), indicating their expectation that the Israeli side is likely to choose confrontation for almost all perceived levels of similarity with the Palestinians (i.e., for all levels where p s < 0.93), practically indicating an overwhelming expectation of Israeli confrontation. Examining the SGSs’ game types according to the revised Rapoport and Guyer taxonomy (Fischer et al.,2024;Rapoport & Guyer,1966) shows that 70% of Israeli participants’ SGSs are absolutely stable games, while only 40% of Palestinian participants’ SGSs are absolutely stable games. Since absolutely stable games are likely to indicate an expectation of mutual cooperation, this classification of games also points to a critical strategic gap, where many more Israeli participants (than Palestinian participants) expected a cooperative interaction between the parties. Notice that almost none of the participants tended to perceive the Israeli–Palestinian conflict as a PD-like game, as already suggested for ecologically valid conflicts (Johnson et al.,2002;Northcott & Alexandrova,2015), and specifically for the Israeli–Palestinian conflict (Rowley & Webb,2007). Classifying SGSs according to the SERS-based taxonomy shows that 64% of the Israeli SGSs are similarity-sensitive for the Israeli side, and 76% of the Palestinian SGSs are similarity-sensitive for the Palestinian side. These relatively high proportions highlight the important role of similarity perceptions in determining the conflict’s trajectory. Examining the elicited similarity perceptions reported by the Israeli participants reveals a rather homogeneous distribution, where all similarity levels (apart from extremely high values) are equally represented. In contrast, the distribution of Palestinian similarity perceptions is characterized by a large proportion, about 40%, of extremely low similarity perceptions, ranging from 0 to 10. This indicates that many Palestinian participants are likely to prefer the choice of a confrontational alternative. Calculating Israeli and Palestinian SERS-based EVs and assuming individuals choose the alternative with the higher EV reveals a higher proportion of cooperative choices (66%) for the Israeli side than for the Palestinian side (49%). In other words, according to SERS, Israelis were somewhat more likely than the Palestinians to choose a cooperative alternative. Further splitting participants’ samples into quartiles, according to the reported intergroup similarity perceptions, and calculating the average SGS similarity thresholds for each quartile, shows that the lower the similarity perception with the other party, the higher the similarity threshold of the game, both increasing the motivation to choose confrontational alternatives. As shown, Israeli participants in three reported similarity quartiles (all but the lowest similarity quartile) were motivated to choose a cooperative alternative, whereas Palestinian participants in three similarity quartiles (all but the highest similarity quartile) were motivated to choose a confrontational alternative. Overall, the findings indicate differences in subjective perceptions of Israeli and Palestinian participants. Israeli attitudes tended to lean towards cooperation, while Palestinian attitudes exhibit more ambivalence and a greater inclination towards confrontational expectations from both parties. Since Israeli and Palestinian participants were sampled well before the 7 October 2023 atrocities and the following war, the reported subjective attitudes help elucidate the preconceptions that might have contributed to the outbreak of hostilities and to the Israeli misperception of Palestinian attitudes. Clearly the outbreak of war is likely to have modified the perceptions of both parties; therefore, the elicited SGSs may Games 2025,16, 3 21 of 23 no longer provide adequate representations of the current parties’ strategic perceptions. Nonetheless, the results emphasize the critical role of similarity perceptions, which seem to have played an important role in determining the parties’ attitudes and expectations, so that even participants with rather cooperative perceptions of the conflict (those who provided SGSs with relatively low similarity thresholds) were still expected to prefer confrontational alternatives, due to their low similarity perceptions. This suggests that peace-building efforts should focus on enhancing perceptions of inter-group similarity. While challenging, this may be more achievable than altering actual or individually perceived payoff structures. Although neither shifting similarity perceptions nor modifying actual or perceived payoffs is easy, both strategies offer pathways to foster cooperation and peaceful interactions in the Israeli–Palestinian conflict as well as in other social and political conflicts. Finally, two cautious remarks are warranted. (i) The study relied on a convenience sample. Consequently, the two groups differ in terms of participants’ average age. This difference suggests that some of the observed differences may be attributed to the higher average age of the Palestinian sample. While Peterson et al. (2020) find that political attitudes are remarkably stable over the long term, they also note that, when changes do occur, they are more likely to reflect shifts from liberal to conservative attitudes. This implies that younger participants are likely to display more liberal attitudes, which could be associated with a higher proportion of absolutely stable games, lower similarity thresholds, and higher values of elicited similarity perceptions. (ii) Clearly, using two-by-two games to model ecologically valid conflicts may oversimplify the strategic dynamics involved. However, this modeling approach focuses participants on critical choices, highlights the differences between the parties, and provides a basic, straightforward representation of the conflict and its possible resolutions. To conclude, we wish to emphasize the benefits of shifting from externally imposed game matrices to subjectively perceived interaction structures, as represented by SGSs. This approach allows for the analysis of the interactions people perceive and respond to, rather than the games they are expected to be playing. Supplementary Materials: The following supporting information can be downloaded at: https:// www.mdpi.com/article/10.3390/g16010003/s1, Figure S1. Examples of the three classes of the SERS-based taxonomy of games, References Loomes and Sugden (1982), Nash (1950) and Rapoport and Guyer (1966) are cited in the Supplementary Materials. Author Contributions: Conceptualization, I.F.; Methodology, I.F., S.A. and L.G.; Software, L.G.; Formal analysis, S.A. and L.G.; Investigation, I.F.; Data curation, S.A.; Writing—original draft preparation, I.F. and S.A.; Writing—review and editing, I.F. and S.A.; Supervision, I.F.; Funding acquisition, I.F. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Israel Science Foundation, grant number 64015. Data Availability Statement: All data are available on the Open Science Framework (OSF) https:// osf.io/xwrud/?view_only=db1913c30f3a42108e838e95f2d20e12 (accessed on 11 July 2024). Conflicts of Interest: The authors declare no conflicts of interest. Notes 1 Note that while the classic PD game requires a strict order of the payoffs, T > R > P > S (Flood,1958), in the category of PD-like games we allow T = R for one of the players. 2 A cell which no player is motivated to leave unilaterally while assuming the other player does not change his choice (i.e., a Nash equilibrium), yet there exists another cell in which both players obtain higher payoffs (and therefore the examined cell is not a Pareto equilibrium). Games 2025,16, 3 22 of 23 3 Between session reliability for participants who were judged to be alignable across sessions was 7.08 (out of 10), SD = 1.87. See Fischer et al. (2024) for the computation of Matrices Strategic Resemblance Index (MSRI) applied here for assessing between session reliability. References Ames, D. R., Weber, E. U., & Zou, X. (2012). Mind-reading in strategic interaction: The impact of perceived similarity on projection and stereotyping. Organizational Behavior and Human Decision Processes,117(1), 96–110. [CrossRef] Axelrod, R. M. (1984). The evolution of cooperation. Basic Books. Banaji, M. R. (2002). Stereotypes, social psychology. In N. J. Smelser, & P. B. Baltes (Eds.), International encyclopedia of the social and behavioral sciences (pp. 15100–15104). Elsevier. Bornstein, G., & Ben-Yossef, M. (1994). Cooperation in intergroup and single-group social dilemmas. Journal of Experimental Social Psychology,30(1), 52–67. [CrossRef] Brams, S. (1993). Theory of moves. American Scientist,81(6), 562–570. [CrossRef] Budescu, D. V., & Rapoport, A. (1994). Subjective randomization in one-and two-person games. Journal of Behavioral Decision Making, 7(4), 261–278. [CrossRef] Chierchia, G., & Coricelli, G. (2015). The impact of perceived similarity on tacit coordination: Propensity for matching and aversion to decoupling choices. Frontiers in Behavioral Neuroscience,9(202), 1–13. [CrossRef] [PubMed] Fischer, I. (2009). Friend or foe: Subjective expected relative similarity as a determinant of cooperation. Journal of Experimental Psychology: General,138(3), 341–350. [CrossRef] Fischer, I. (2012). Similarity or reciprocity? On the determinants of cooperation in similarity-sensitive games. Psychological Inquiry, 23(1), 48–54. [CrossRef] Fischer, I., & Avrashi, S. (2024). Identifying optimal decision-making strategies and determining effective messaging to maximize the expected outcomes of potential human–extraterrestrial encounters. International Journal of Astrobiology,23, e21. [CrossRef] Fischer, I., Avrashi, S., & Givon, L. (2024). Subjective game structures: Eliciting alternatives and payoffs to study the properties of social interactions. Experimental Economics,27, 922–944. [CrossRef] Fischer, I., Frid, A., Goerg, S. J., Levin, S. A., Rubenstein, D. I., & Selten, R. (2013). Fusing enacted and expected mimicry generates a winning strategy that promotes the evolution of cooperation. National Acad Sciences,110, 10229–10233. [CrossRef] Fischer, I., Levin, S. A., Rubenstein, D. I., Avrashi, S., Givon, L., & Oz, T. (2022). Interacting with others while reacting to the environment. Behavioral and Brain Sciences,45(E106), 27–28. [CrossRef] [PubMed] Fischer, I., & Savranevski, L. (2023). The effect of similarity perceptions on human cooperation and confrontation. Scientific Reports, 13(1), 19849. [CrossRef] [PubMed] Flood, M. (1958). Some experimental games. Management Science,5(1), 5–26. [CrossRef] Halevy, N., Chou, E. Y., & Murnighan, J. K. (2012). Mind games: The mental representation of conflict. Journal of Personality and Social Psychology,102(1), 132–148. [CrossRef] [PubMed] Hardin, G. (1968). The tragedy of the commons. Science,162(3859), 1243–1248. [CrossRef] [PubMed] Johnson, D. D. P., Stopka, P., & Bell, J. (2002). Individual variation evades the prisoner’s dilemma. BMC Evolutionary Biology,2(1), 15. [CrossRef] [PubMed] Loomes, G., & Sugden, R. (1982). Regret theory: An alternative theory of rational choice under uncertainty. The Economic Journal, 92(368), 805. [CrossRef] Nash, J. F. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences,36(1), 48–49. [CrossRef] [PubMed] Northcott, R., & Alexandrova, A. (2015). Prisoner’s dilemma doesn’t explain much. In P. Martin (Ed.), The prisoner’s dilemma. Classic philosophical arguments (pp. 64–84). Cambridge University Press. Available online: https://philpapers.org/rec/NORPDD (accessed on 28 December 2022). Ostrom, E. (1998). A Behavioral approach to the rational choice theory of collective action: Presidential address, American political science association, 1997. American Political Science Review,92(1), 1–22. [CrossRef] Peterson, J. C., Smith, K. B., & Hibbing, J. R. (2020). Do people really become more conservative as they age? The Journal of Politics, 82(2), 600–611. [CrossRef] Plous, S. (1985). Perceptual illusions and military realities. Journal of Conflict Resolution,29(3), 363–389. [CrossRef] Rapoport, A. (1967). Exploiter, leader, hero, and martyr: The four archetypes of the 2 × 2 game. Behavioral Science,12(2), 81–84. [CrossRef] [PubMed] Rapoport, A., & Chammah, A. M. (1965). Prisoner’s dilemma: A study in conflict and cooperation. University of Michigan Press. Rapoport, A., & Chammah, A. M. (1966). The game of chicken. American Behavioral Scientist,10(3), 10–28. [CrossRef] Games 2025,16, 3 23 of 23 Rapoport, A., & Guyer, M. (1966). A taxonomy of 2 ×2 games. General Systems,11, 203–214. Rowley, C. K., & Webb, M. J. (2007). Israel and Palestine: The slow road to peace or the fast track to mutual annihilation? Public Choice, 132(1–2), 7–26. [CrossRef] Skyrms, B. (2001). The stag hunt. Proceedings and Addresses of the American Philosophical Association,75(2), 31. [CrossRef] Toma, C., Corneille, O., & Yzerbyt, V. (2012). Holding a mirror up to the self. Personality and Social Psychology Bulletin,38(10), 1259–1271. [CrossRef] von Neumann, J., & Morgenstern, O. (2007). Theory of games and economic behavior (60th anniversary commemorative edition). Princeton University Press. [CrossRef] Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.