Best experienced payoff dynamics and cooperation in the centipede game
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Sandholm, William H.; Izquierdo, Segismundo S.; Izquierdo, Luis R. Article Best experienced payoff dynamics and cooperation in the centipede game Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Sandholm, William H.; Izquierdo, Segismundo S.; Izquierdo, Luis R. (2019) : Best experienced payoff dynamics and cooperation in the centipede game, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 14, Iss. 4, pp. 1347-1385, https://doi.org/10.3982/TE3565 This Version is available at: https://hdl.handle.net/10419/217105 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-nc/4.0/
Theoretical Economics 14 (2019), 1347–1385 1555-7561/20191347 Best experienced payoff dynamics and cooperation in the centipede game William H. Sandholm Department of Economics, University of Wisconsin Segismundo S. Izquierdo BioEcoUva, Department of Industrial Organization, Universidad de Valladolid Luis R. Izquierdo Department of Civil Engineering, Universidad de Burgos We study population game dynamics under which each revising agent tests each of his strategies a fixed number of times, with each play of each strategy being against a newly drawn opponent, and chooses the strategy whose total payoff was highest. In the centipede game, these best experienced payoff dynamics lead to cooperative play. When strategies are tested once, play at the almost globally stable state is concentrated on the last few nodes of the game, with the proportions of agents playing each strategy being largely independent of the length of the game. Testing strategies many times leads to cyclical play. Keywords. Evolutionary game theory, backward induction, centipede game, computational algebra. JEL classification. C72, C73. 1. Introduction The discrepancy between the conclusions of backward induction reasoning and observed behavior in certain canonical extensive form games is a basic puzzle of game theory. The centipede game (Rosenthal (1981)), the finitely repeated prisoner’s dilemma, and related examples can be viewed as models of relationships in which each participant has repeated opportunities to take costly actions that benefit his partner and in which there is a commonly known date at which the interaction will end. Experimental and anecdotal evidence suggests that cooperative behavior may persist until close to William H. Sandholm: [email protected] Segismundo S. Izquierdo: [email protected] Luis R. Izquierdo: [email protected] We thank Dan Friedman, Drew Fudenberg, Ken Judd, Panayotis Mertikopoulos, Erik Mohlin, Ignacio Monzón, Arthur Robson, Marzena Rostek, Ariel Rubinstein, Larry Samuelson, Ryoji Sawa, Andy Skrzypacz, Lones Smith, Mark Voorneveld, Marek Weretka, and, especially, Antonio Penta for helpful discussions and comments. Financial support from the U.S. National Science Foundation (Grants SES-1458992 and SES- 1728853), the U.S. Army Research Office (Grants W911NF-17-1-0134 MSN201957), Project ECO2017-83147- C2-2-P (MINECO/AEI/FEDER, UE), and the Spanish Ministerio de Educación, Cultura, y Deporte (Grants PRX15/00362 and PRX16/00048) is gratefully acknowledged. ©2019 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://econtheory.org.https://doi.org/10.3982/TE3565
1348 Sandholm, Izquierdo, and Izquierdo Theoretical Economics 14 (2019) the exogenous terminal date (McKelvey and Palfrey (1992)). But the logic of backward induction leads to the conclusion that there will be no cooperation at all. Work on epistemic foundations provides room for wariness about unflinching appeals to backward induction. To support this prediction, one must assume that there is always common belief that all players will act as payoff maximizers at all points in the future, even when many rounds of previous choices argue against such beliefs.1Thus, the simplicity of backward induction belies the strength of the assumptions needed to justify it, and this strength may help explain why backward induction does not yield descriptively accurate predictions in some classes of games.2 This paper studies a dynamic model of behavior in games that maintains the assumption that agents respond optimally to the information they possess. But rather than imposing strong assumptions about agents’ knowledge of opponents’ intentions, we suppose instead that agents’ information comes from direct but incomplete experience with playing the strategies available to them. As with the earlier work of Osborne and Rubinstein (1998)andSethi (2000), our model is best viewed not as one that incorporates irrational choices, but rather as one of rational choice under particular restrictions on what agents know. Following the standard approach of evolutionary game theory, we suppose that two populations of agents are recurrently randomly matched to play a two-player game. This framework accords with some experimental protocols, and can be understood more broadly as a model of the formation of social norms (Young (1998)). At random times, each agent receives opportunities to switch strategies. At these moments the agent plays each of his strategies against κopponents drawn at random from the opposing population, with each play of each strategy being against a newly drawn opponent. He then switches to the strategy that achieved the highest total payoff, breaking ties in favor of the lowest-numbered strategy. Standard results imply that when the populations are large, the agents’ aggregate behavior evolves in an essentially deterministic fashion, obeying a differential equation that describes the expected motion of the stochastic process described above (Benaïm and Weibull (2003)). We study the properties of this differential equation when agents play the centipede game. Our model builds on earlier work on games played by “procedurally rational players.” If we replace our tie-breaking rule with uniform tie-breaking, then the rest points of the process (with κ=k) would correspond to the S(k) equilibria of Osborne and Rubinstein (1998). The corresponding dynamics were studied by Sethi (2000). These and other dynamics are instances of the broader family of best experienced payoff dynamics (BEP dynamics for short; Sandholm et al. (2019)), which allow for variation in how ties are resolved and in the selection of sets of candidate strategies considered by revising agents. The results we present here are robust to many different model specifications within the family of BEP dynamics. 1For formal analyses, see Binmore (1987), Reny (1992), Stalnaker (1996), Ben-Porath (1997), Halpern (2001), and Perea (2014). 2As an alternative, one could apply Nash equilibrium, which also predicts noncooperative behavior in the games mentioned above, but doing so replaces assumptions about future rationality with the assumption of equilibrium knowledge, which may not be particularly more appealing; see Dekel and Gul (1997).
Theoretical Economics 14 (2019) Dynamics and cooperation in centipede 1349 Our analysis of best experienced payoff dynamics in the centipede game uses techniques from dynamical systems theory. What is more novel is our reliance on algorithms from computational algebra and perturbation bounds from linear algebra, which allow us to solve exactly for the rest points of our differential equations and to perform rigorous stability analyses in centipede games with up to six decision nodes. We complement this approach with numerical analyses of cases in which analytical results cannot be obtained. Our initial results focus on dynamics under which each tested strategy is tested exactly once (κ=1), so that agents’ choices depend only on ordinal properties of payoffs. In centipede games, under the BEP dynamics studied here, the backward induction state—the state at which all agents in both populations stop at their first opportunity— is a rest point. However, we prove that this rest point is always repelling: the appearance of agents in either population who cooperate to any degree is self-reinforcing and eventually causes the backward induction solution to break down completely. We next obtain strong lower bounds on the total weight placed on cooperative strategies at any other rest points of the BEP dynamic. At any such rest point, the probability that play during a random match leads to one of the last five terminal nodes is above 096, and the probability that play leads to one of the last seven terminal nodes is virtually 1. We then use tools from computational algebra to perform an exact analysis of games with up to six decision nodes, and we perform numerical analyses of longer games. In all cases, we find that besides the unstable backward induction state, the dynamics have exactly one other rest point.3The form of this rest point is essentially independent of the length of the game. The rest point has virtually all players choosing to continue until the last few nodes of the game. Moreover, this rest point is dynamically stable, attracting solutions from all initial conditions other than the backward induction state. Thus if agents make choices based on experienced payoffs, testing each strategy once and choosing the one that performed best, then play converges to a stable rest point that exhibits high levels of cooperation. To explain why, we first observe that cooperative strategies are most disadvantaged when they are most rare—specifically, in the vicinity of the backward induction state. Near this state, the most cooperative agents would obtain higher expected payoffs by stopping earlier. However, when an agent considers switching strategies, he tests each of his strategies against new, independently drawn opponents. He may thus test a cooperative strategy against a cooperative opponent, and test less cooperative strategies against less cooperative opponents, in which case his best experienced payoff will come from the cooperative strategy. Our analysis confirms that this possibility indeed leads to instability.4After this initial entry, the high payoffs generated by cooperative strategies 3While traditional equilibrium notions in economics require stasis of choice, interior rest points of population dynamics represent situations in which individuals’ choices fluctuate even as the expected change in aggregate behavior is null; see Section 2.2. 4Specifically, linearizing any given specification of the dynamics at the backward induction state identifies a single eigenvector with a positive eigenvalue (Appendix A). This eigenvector describes the mixture of strategies in the two populations whose entry is self-reinforcing and identifies the direction toward which all other disturbances of the backward induction state are drawn. Direct examination of the dynamics provides a straightforward explanation of why the given mixture of entrants is successful (Example 2).
1350 Sandholm, Izquierdo, and Izquierdo Theoretical Economics 14 (2019) when matched against one another spurs their continued growth. This growth is only abated when virtually all agents are choosing among the most cooperative strategies. Our final results consider the effects of the number of trials κof each strategy during testing on predictions of play. It seems clear that if the number of trials is made sufficiently large, so that the agents’ information about opponents’ behavior is quite accurate, then the population’s behavior should come to resemble a Nash equilibrium. Indeed, when agents possess exact information, so that aggregate behavior evolves according to the best response dynamic (Gilboa and Matsui (1991), Hofbauer (1995)), the results of Xu (2016) imply that every solution trajectory converges to the set of Nash equilibria, all of which entail stopping at the initial node. Our analysis shows, however, that stable cooperative behavior can persist even for substantial numbers of trials. To start, we prove that the backward induction state is unstable as long as the number of trials κis less than the length of the game. For a larger number of trials, the backward induction state becomes locally stable, but numerical evidence suggests that its basin of attraction is very small. Examining centipede games of length d=4in detail, we find that a unique, attracting interior rest point with substantial cooperation persists for moderate numbers of trials. With many trials, numerical analysis suggests the attractor is always a single cycle that includes significant amounts of cooperation for numbers of trials as large as 200. We discuss in Section 4 how the robustness of cooperation to fairly large numbers of trials can be explained using simple central limit theorem arguments. Our main technical contribution lies in the use of methods from computational algebra and perturbation theorems from linear algebra to prove results about the properties of our dynamics. The starting point for this analysis—one that suggests a broader scope for our approach—is that decision procedures based on sampling from a population are described by multivariate polynomials with rational coefficients. In particular, BEP dynamics are described by systems of such equations, so finding their rest points amounts to finding the zeros of these polynomial systems. To accomplish this, we compute a Gröbner basis for the set of polynomials that defines each instance of our dynamics; this new set of polynomials has the same zeros as the original set, but its zeros can be computed by finding the roots of a single (possibly high-degree) univariate polynomial.5Exact representations of these roots, known as algebraic numbers, can then be obtained by factoring the polynomial into irreducible components and then using algorithms based on classical results to isolate each component’s real roots.6With these exact solutions in hand, we can rigorously assess the rest points’ local stability through a linearization analysis. So as to obviate certain intractable exact calculations, this analysis takes advantage of both an eigenvalue perturbation theorem and a bound on the condition number of a matrix that does not require the computation of its inverse. The code used to obtain the exact and numerical results is available as a Mathematica notebook posted on GitHub and on the authors’ websites. The Supplemental Material provides background and details about both the exact and the numerical analyses, and reports certain numerical results in full detail. 5See Buchberger (1965) and Cox et al. (2015). For applications of Gröbner bases in economics, see Kubler et al. (2014). 6See von zur Gathen and Gerhard (2013), McNamee (2007), and Akritas (2010).
Theoretical Economics 14 (2019) Dynamics and cooperation in centipede 1351 Related literature Previous work relating backward induction and deterministic evolutionary dynamics has focused on the replicator dynamic of Taylor and Jonker (1978) and the best response dynamic of Gilboa and Matsui (1991)andHofbauer (1995). Cressman and Schlag (1998) (see also Cressman (1996,2003)) show that in generic perfect information games, every interior solution trajectory of the replicator dynamic converges to a Nash equilibrium. Likewise, Xu (2016)(seealsoCressman (2003)) shows that in such games, every solution trajectory of the best response dynamic converges to a component of Nash equilibria. In both cases, the Nash equilibria approached need not be subgame perfect and the Nash equilibrium components generally are not locally stable. Focusing on the centipede game with three decision nodes, Ponti (2000) shows numerically that perturbed versions of the replicator dynamic exhibit cyclical behavior, with trajectories approaching and then moving away from the Nash component. In contrast, we show that for small and moderate numbers of tests, best experienced payoff dynamics lead to a stable distribution of cooperative strategies far from the Nash component. Osborne and Rubinstein’s (1998)notionofS(k) equilibrium corresponds to the rest points of the BEP dynamic under which agents test all strategies, subject each to ktrials, and break ties via uniform randomization.7While most of their analysis focuses on simultaneous move games, they show that in centipede games, the probability with which player 1stops immediately in any S(1)equilibrium must vanish as the length of the game grows large. As we soon observe (Observation 1), this conclusion may fail if uniform tie-breaking is not assumed, with the backward induction state being an equilibrium. Nevertheless, more detailed analyses below show that this equilibrium state is unstable under BEP dynamics. Building on Osborne and Rubinstein (1998), Sethi (2000) introduces BEP dynamics under which all strategies are tested and ties are broken uniformly.8He shows that both dominant strategy equilibria and strict equilibria can be unstable under these dynamics, while dominated strategies can be played in stable equilibria. The latter fact is a basic component of our analysis of cooperative behavior. Berkemer (2008) considers the local stability of the unique rationalizable strategy profile in the traveler’s dilemma of Basu (1994)underSethi’s (2000) dynamics, obtaining a sufficient condition for the instability of the rationalizable state. He shows numerically that the stable S(1)equilibrium becomes independent of the number of strategies in the game and he provides evidence from agent-based simulations that larger numbers of trials during testing can lead to cyclical behavior.9 Earlier efforts to explain cooperative behavior in centipede and related games have followed a different approach, applying equilibrium analyses to augmented versions of the game. The best known example of this approach is the work of Kreps et al. (1982). 7For extensions of S(k) equilibrium to more complex testing procedures, see Rustichini (2003). 8Cárdenas et al. (2015) and Mantilla et al. (2019) use these dynamics to explain stable non-Nash behavior in public goods games. 9For complementary models of dynamics based on a single sample, see Sandholm (2001), Kosfeld et al. (2002), Droste et al. (2003), and Oyama et al. (2015).
1352 Sandholm, Izquierdo, and Izquierdo Theoretical Economics 14 (2019) These authors modify the finitely repeated prisoner’s dilemma by assuming that one player attaches some probability to his opponent having a fixed preference for cooperative play. They show that in all sequential equilibria of long enough versions of the resulting Bayesian game, both players act cooperatively for a large number of initial rounds.10 To justify this approach, one must assume that the augmentation of the original game is commonly understood by the players, that the players act in accordance with a rather complicated equilibrium construction, and that the equilibrium knowledge assumptions required to justify sequential equilibrium apply. In contrast, our model makes no changes to the original game other than placing it in a population setting, and it is built upon the assumption that agents’ choices are optimal given their experiences during play. 2. Best experienced payoff dynamics in the centipede game 2.1 Normal form games and population games A two-player normal form game G={(S1S2) (A B)}is defined by pairs of strategy sets Sp={1sp}and payoff matrices AB ∈Rsp×sq,p q ∈{12},p=q.EntriesAij and Bij represent the two players’ payoffs when strategy profile (ij) ∈S1×S2is played. When considering extensive form games, our analysis focuses on the reduced normal form, whose strategies specify an agent’s “plan of action” for the game, but not his choices at decision nodes that are ruled out by his own previous choices. In our population model, members of two unit-mass populations are matched to play a two-player game. A population state for population 1 is an element of X={x∈ Rs1 +:i∈S1xi=1},wherexiis the fraction of population 1 players choosing strategy i. Likewise Y={y∈Rs2 +:i∈S2yi=1}is the set of population states for population 2. Thus, xand yare formally equivalent to mixed strategies for players 1 and 2, and elements of the set =X×Yare formally equivalent to mixed strategy profiles. In a slight abuse of terminology, we also refer to elements of as population states. 2.2 Revision protocols and evolutionary dynamics To define evolutionary game dynamics, we follow the standard approach of specifying microfoundations in terms of revision protocols.11 We suppose that at all times t∈ [0∞), each agent has a strategy he uses when matched to play game G.Theempirical distributions of these strategies are described by the population state ξ(t) =(x(t) y(t)). 10McKelvey and Palfrey (1992) show that this analysis extends to the centipede game. A different augmentation is considered by Jehiel (2005), who assumes that agents bundle decision nodes from contiguous stages into analogy classes and view the choices at all nodes in a class interchangeably. Alternatively, following Radner (1980), one can consider versions of centipede in which the stakes of each move are small and analyze these games using εequilibrium; see Friedman and Oprea (2012) for a discussion. But as Binmore (1998) observes, the existence of a non-Nash εequilibrium depends on the relative sizes of the stakes and of ε, and the backward induction solution always persists as a Nash equilibrium and, hence, as an ε equilibrium. 11See Björnerstedt and Weibull (1996), Weibull (1995), Sandholm (2010a,b,2015), and Izquierdo et al. (2018).
Theoretical Economics 14 (2019) Dynamics and cooperation in centipede 1353 Agents occasionally receive opportunities to switch strategies according to independent rate 1Poisson processes. An agent who receives an opportunity considers switching to a new strategy, making his decision by applying a revision protocol. Formally, arevision protocol for population 1is described by a map (Ay) →σ1(Ay) ∈Xs1that assigns own payoff matrices and opposing population states to matrices of conditional switch probabilities,whereσ1 ij (Ay) is the probability that an agent playing strategy i∈S1who receives a revision opportunity switches to strategy j∈S1. Likewise, a revision protocol for population 2is described by a map (Bx) →σ2(Bx) ∈Ys2with an analogous interpretation.12 It is well known that if the population sizes are large, the Markov process implicitly defined by the above procedure is well approximated by solutions to a differential equation defined by the expected motion of the process (Benaïm and Weibull (2003)). Here this differential equation takes the form ˙ xi= j∈S1 xjσ1 ji(Ay) −xifor all i∈S1 ˙ yi= j∈S2 yjσ2 ji(Bx) −yifor all i∈S2 (1) Equation (1) is easy to interpret. Since revision opportunities are assigned to agents randomly, there is an outflow from each strategy iproportional to its current level of use. To generate inflow to i, an agent playing some strategy jmust receive a revision opportunity, and applying his revision protocol must lead him to play strategy i. Outside of monomorphic (i.e., pure) cases, the rest points of the dynamic (1)should not be understood as equilibria in the traditional game-theoretic sense. Rather, they represent situations in which agents perpetually switch among strategies, but with the expected change in the use of each strategy equaling zero.13 At states that are locally stable under the dynamic (1), fluctuations in any direction are generally undone by the action of (1) itself. Contrariwise, fluctuations away from unstable equilibria are reinforced, so we should not expect such states to be observed. 2.3 Best experienced payoff protocols and dynamics We now introduce the class of revision protocols and dynamics that we study in this paper. A best experienced payoff protocol is defined by a triple (τκβ)consisting of a test set rule τ,anumber of trials κ,andatie-breaking rule β. The triple (τκβ) defines a revision protocol in the following way. When an agent currently using strategy i∈Sp receives an opportunity to switch strategies, he draws a set of strategies Rp⊆Spto test according to the distribution τpon the collection of subsets of Spwith at least two elements. He then plays each strategy in Rpin κrandom matches against members of 12When σ1 ij and σ2 ij are independent of the current strategy i,asistrueforthedynamic(3) we focus on here, it is equivalent to interpret the process as one in which agents play a fixed strategy until leaving the population, when they are replaced by new agents whose strategies are determined by applying σ1and σ2. 13Thus in the finite-population version of the model, variations in the use of each strategy would be observed. For a formal analysis, see Sandholm (2003).
1354 Sandholm, Izquierdo, and Izquierdo Theoretical Economics 14 (2019) the opposing population. He thus engages in #Rp×κrandom matches in total, facing distinct sets of opponents when testing different strategies. The agent then selects the strategy in Rpthat earned him the highest total payoff, breaking ties according to rule β. The triple (τκβ)thus defines a revision protocol σpfor each population p. Inserting these revision protocols into (1)definesabest experienced payoff dynamic. Our analysis here focuses on the test-set rule test-all,τall, under which a revising agent tests all of his strategies, and on the tie-breaking rule min-if-tie,βmin,which chooses the lowest-numbered optimal strategy. We refer to the resulting dynamics (1) as BEP(τallκβmin)dynamics. BEP dynamics based on other specifications of test-set and tie-breaking rules are studied in a companion paper (Sandholm et al. (2019)); they are also discussed briefly in Section 3.4. Importantly, if we retain τall,butreplaceβmin with uniform tie-breaking, then the rest points of the dynamic (1)aretheS(k) equilibria of Osborne and Rubinstein (1998)(withk=κ), and the dynamic itself is the one studied by Sethi (2000). In extensive form games like centipede, different strategies often earn the same payoffs, so the choice of tie-breaking rule matters. Because of our convention for numbering strategies in the centipede game (see below), the min-if-tie rule will be the one that is least conducive to cooperative play. Choice probabilities under best experienced payoff dynamics depend only on the payoffs strategies earn during testing; they do not require agents to track the choices made by their opponents. This property makes the dynamics appealing as a simple model of play for extensive form games. Typically, a single play of an extensive form game does not reveal the strategy chosen by one’s opponent, but only the portion of that strategy required to determine the path of play. Consequently, it is not straightforward to specify how agents should use their experience of play to assess opponents’ choices of strategies. Because they focus on the performances of own strategies, best experienced payoff dynamics avoid such ambiguities. 2.4 The centipede game Centipede (Rosenthal (1981)) is a two-player extensive form game with d≥2decision nodes (Figure 1). Each node presents two actions: stop and continue. The nodes are arranged linearly, with the first one assigned to player 1 and subsequent nodes assigned in an alternating fashion. A player who stops ends the game. A player who continues suffers a cost of 1 but benefits his opponent 3, and sends the game to the next decision node if one exists. Figure 1. The centipede game of length d=8.
Theoretical Economics 14 (2019) Dynamics and cooperation in centipede 1361 Now, applying (6)and(7) sequentially yields the inequalities ¯ x[1]≥1−(1−¯ y[1])2≥1−(1−r)2>06752 ¯ y[2]≥1−(1−¯ x[1])3≥1−(1−r)6>09657 ¯ x[2]≥1−(1−¯ y[2])3≥1−(1−r)18 >1−10−4 ¯ y[3]≥1−(1−¯ x[2])4≥1−(1−r)72 >1−10−16 (11) Part of the intuition behind the proof of Proposition 3 is straightforward. Consider an interior rest point ξ=(xy) of the BEP(τall1βmin)dynamic when dis even. Inequality (6)saysthatif1−¯ y[k]is small (i.e., if few population 2 players stop before [k]), then 1−¯ x[k]is even smaller (i.e., few population 1 players stop before [k]), since some test of a more cooperative strategy will very likely lead to a high payoff. Likewise, by (7), if 1−¯ x[k]is small, then 1−¯ y[k+1]is smaller still. These inequalities imply that any lower bound on ¯ y[k]will quickly propagate into much stronger lower bounds on ¯ x[k],¯ y[k+1], ¯ x[k+1],andsoon(see(11)). Initiating this chain of reasoning requires a less obvious step: we combine inequality (7) with a weak bound (9)on1−¯ x[k]that depends on ¯ y[k+1] alone. This combination gives us the initial inequality ¯ y[1]≥04301, which in turn leads to the strong bounds stated in the proposition. 3.2 Results based on exact computations Proposition 3 places strong lower bounds on the degree of cooperation arising at any rest point other than the unstable backward induction state. To gain a more precise understanding of the form and stability of such rest points, we turn to exact computations. Because the dynamic (3) is a system of polynomials with rational coefficients, its zeros can in principle be found by computing a Gröbner basis for the system. The Gröbner basis is a new system of equations that has the same zeros as the original system, but can be solved by backward substitution. Once the Gröbner basis has been obtained, polynomial factoring and root finding algorithms can be used to identify its zeros and, hence, the zeros of the original system. Applying these techniques, which are described in detail in Appendix C, leads to part (i) of the following result. Proposition 4. In centipede games of lengths 3≤d≤6, the following statements hold: (i) The BEP(τall1βmin)dynamic has exactly two rest points: ξ†,andξ∗=ξ∗(d) ∈ int(). (ii) The rest point ξ∗is asymptotically stable. Exact solutions can only be obtained for games of length at most 6because of the computational demands of computing the Gröbner bases. Two indications of these demands are that when d=6, the leading (univariate) polynomial from the Gröbner basis is of degree 221 and a coefficient of one of the polynomials in the basis has 13,278 digits. Table 1 reports the approximate values of the interior rest points ξ∗=ξ∗(d), referring to strategies using the last-to-first notation [k]introduced in Section 2.4. Evidently, the
1362 Sandholm, Izquierdo, and Izquierdo Theoretical Economics 14 (2019) Population pPopulation q [3] [2] [1] [0] [3] [2] [1] [0] d=30618034 0381966 0381966 0381966 0236068 d=40113625 0501712 0384663 0337084 0419741 0243175 d=50113493 0501849 0384658 0001462 0335672 0419706 0243160 d=6312 ×10−90113493 0501849 0384658 0001462 0335672 0419706 0243160 Note: The pdenotes the owner of the penultimate decision node; the qdenotes the owner of the last decision node. Table 1. “Exact” interior rest points ξ∗=ξ∗(d) of the BEP(τall1βmin)dynamic. d=3−1±03820 −1 d=4−11411 ±03277i−08589 ±03277i d=5−11355 ±03284i−08645 ±03284i−1 d=6−11355 ±03284i−08645 ±03284i−1±974 ×10−5i Table 2. Eigenvalues of the derivative matrices DV (ξ∗)of the BEP(τall1βmin)dynamic. masses on each strategy are nearly identical for games of lengths 4,5,and6, with nearly all of the weight in both populations being placed on continuing to the end, stopping at the last node, or stopping at the penultimate node. In principle, it is possible to prove the local stability of the rest points ξ∗=ξ∗(d) using linearization. But since the components of ξ∗are algebraic numbers, computing the eigenvalues of DV (ξ∗)requires finding the exact roots of a polynomial with algebraic coefficients, a computationally intensive problem. Fortunately, we can prove local stability without doing so. Instead, we compute the eigenvalues of the matrix DV (ξ),where ξis a rational point that is very close to ξ∗, showing that these eigenvalues all have negative real part. Proposition 6 in Appendix D establishes an upper bound on the distances between the eigenvalues of DV (ξ) and DV (ξ∗). Importantly, this bound can be evaluated without having to compute the roots of a polynomial with algebraic coefficients or to invert a matrix with algebraic components, as both of these operations quickly become computationally infeasible. Combining these steps allows us to conclude that the eigenvalues of DV (ξ∗)also have negative real part. For a detailed presentation of this argument, see Appendix D. The approximate eigenvalues of DV (ξ∗)are reported in Table 2. Note that the eigenvalues for games of length 5and 6are nearly identical, with the replacement of an eigenvalue of −1by a pair of complex eigenvalues that are very close to −1. 3.3 Numerical results Because exact methods allow us to determine only the rest points of the BEP(τall1 βmin)dynamic in centipede games of lengths d≤6, we use numerical methods to study games of lengths 7–20. We know from Proposition 3 that at any rest point besides the backward induction state ξ†, the weight on strategies that stop before either player’s
Theoretical Economics 14 (2019) Dynamics and cooperation in centipede 1363 Figure 2. The stable rest point ξ∗=ξ∗(d) of centipede under the BEP(τall1βmin)dynamic for game lengths d=310 and d=20. Stacked bars, from the bottom to the top, represent weights on strategy [0] (continue at all decision nodes), [1] (stop at the last node), [2] (stop at the second-to-last node), etc. The dashed line separates exact (d≤6) and numerical (d≥7)results. third-to-last node is very small. This suggests that the presence of earlier nodes should have little bearing on how the game is played. Our numerical analysis suggests that for game lengths 7≤d≤20, there are exactly two rest points: the backward induction state ξ†and an interior rest point ξ∗=ξ∗(d). As Figure 2 illustrates, the form of the interior rest point follows the pattern from Table 1: regardless of the length of the game, nearly all of the mass is placed on each population’s three most cooperative strategies, and the weights on these strategies are essentially independent of the length of the game. Precise numerical estimates of these rest points are provided in Appendix III available in a supplementary file on the journal website, http://econtheory.org/supp3565/supplement.pdf, as are numerical estimates of the eigenvalues of the derivative matrices DV (ξ∗). The latter are essentially identical to those presented in Table 2 for d=6, with the addition of an eigenvalue of ≈−1for each additional decision node. These numerical results suggest that the conclusions about rest points established analytically for games of lengths d≤6continue to hold for longer games: there are always exactly two rest points: the backward induction state ξ†and a stable interior rest point ξ∗whose form barely varies with the length of the game. The facts that the vertex ξ†is repelling, the interior rest point ξ∗=ξ∗(d) is attracting, and these are the only two rest points give us a strong reason to suspect that state ξ∗ attracts all solutions of the BEP(τall1βmin)dynamic other than the stationary solution at ξ†.17 To argue that ξ∗is almost globally stable, we introduce the candidate Lyapunov 17For there to be other solutions that did not converge to ξ∗without the dynamics having another rest point, the flow of the dynamics would need to have very special topological properties. For instance, in a two-dimensional setting, this could occur if ξ∗were contained in a pair of concentric closed orbits, where the inner orbit is repelling and the outer orbit is attracting.
1364 Sandholm, Izquierdo, and Izquierdo Theoretical Economics 14 (2019) function L(xy) = s1 i=2xi−x∗ i2+ s2 j=2yj−y∗ j2 In words, L(xy) is the squared Euclidean distance of (xy) from (x∗y∗)if the points in the state space are represented in Rdby omitting the first components of xand y. The Gröbner basis techniques used in Section 3.2 are not suitable for establishing that Lis a Lyapunov function. For the centipede game of length d=3, we are able to verify that Lis a Lyapunov function using an algorithm from real algebraic geometry called cylindrical algebraic decomposition (Collins (1975)). However, exact implementations of this algorithm fail to terminate in longer games. We therefore verify numerically that Lis a Lyapunov function. For games of lengths 4–20, we chose one billion (109) points from the state space uniformly at random, and evaluated a floating-point approximation of ˙ Lat each point. In all instances, the approximate version of ˙ Levaluated to a negative number. This numerical procedure covers the state space fairly thoroughly for the game lengths we consider,18 and so provides strong numerical evidence that the interior rest point ξ∗is an almost global attractor. 3.4 Other specifications of the dynamics To test the robustness of the preceding results, we repeat the analyses for other specifications of BEP(τ1β)dynamics. In addition to the test-all rule τall,wealsostudieda test-set rule under which the revising agent considers only his current strategy and one other strategy (τtwo), as well as a rule under which the revising agent considers only his current strategy and one adjacent strategy (τadj). The qualitative behavior under these test-set rules is similar to that under τall. The differences worth mentioning are that stable play is concentrated on a larger number of strategies (e.g., nine strategies in total have mass of at least 001 under BEP(τtwo1βmin)) and that the rate of decay of the weights on strategies that stop earlier is not as severe as under τall. The intuition here is simple. Under test-all, a revising agent will try out all of his most cooperative strategies, providing many opportunities for some such strategy to perform best; if instead only two strategies are tested at a time, the selective pressure against less cooperative strategies is weaker. In addition, we also considered alternative tie-breaking rules: stick/min-if-tie,which chooses the agent’s current strategy if it is optimal and chooses the lowest-numbered strategy otherwise, and uniform-if-tie, which randomizes uniformly among the optimal strategies (as in Osborne and Rubinstein (1998)andSethi (2000)). As we noted in Section 2.5, uniform tie-breaking implies that the backward induction state ξ†is not a rest point, rendering a stability analysis of this rest point unnecessary. In other respects, alternate choices of tie-breaking rules have little qualititative impact on behavior. 18By a standard combinatoric formula, the number of states in a grid in =X×Ywith mesh 1 mis m+s1−1 mm+s2−1 m. Applying this formula shows for a game of length 10 that 109is between the numbers of states in grids in of meshes 1 17 (since 22 172=693,479,556) and 1 18 (since 23 182=1,132,255,201). For a game of length 15, the comparable meshes are 1 10 and 1 11 , and for length 20 are 1 7and 1 8.
Theoretical Economics 14 (2019) Dynamics and cooperation in centipede 1365 In summary, the results presented in previous sections are highly robust to alternative specifications of the dynamics. 4. Larger numbers of trials The analysis thus far has focused on cases in which agents test each strategy in their test sets exactly once. We now examine aggregate behavior when each strategy is subject to larger numbers of trials κ, focusing on BEP(τallκβmin)dynamics. 4.1 Instability and stability of the backward induction state Proposition 1 shows that the backward induction state ξ†is a repellor under the BEP(τallκβmin)dynamic with κ=1. The following proposition shows that ξ†remains unstable as long as the number of trials κis less than the length of the game dand then becomes stable for larger numbers of trials. The statement is complicated slightly by the dependence of the crossover point on whether dis even or odd. Proposition 5. Under the BEP(τallκβmin)dynamic in the centipede game of length d, the backward induction state ξ†is unstable if κ≤2d 2and is asymptotically stable otherwise. Like those of the earlier stability analyses, the proof of Proposition 5,whichispresented in Appendix E, is based on linearization. The key observation is that linearizations of BEP dynamics around pure rest points are driven by match results in which exactly one out of all κspmatch partners plays a strategy different from the equilibrium strategy. We show that if κ≤d−1(for deven) or κ≤d(for dodd), there is enough sensitivity to perturbations of the state to ensure the existence of an unstable manifold through ξ†; however, unlike in the κ=1case, ξ†need not be a repellor. Conversely, if these inequalities are violated, strategy 1∈S1earns the highest total payoff after any matching with exactly one discrepant opponent. This insensitivity of population 1’s behavior to small changes in population 2’s behavior ensures local stability. So as to assess the practical relevance of the stability of the backward induction state for larger numbers of trials, we use numerical analysis to estimate the basin of attraction of ξ†and to determine the position of the interior saddle point of the dynamics, which lies on the manifold separating the basin of ξ†from the basin of the main attractor. We focus for tractability on games of length d=4and numbers of trials κ≤100. Details of these analyses are presented in Appendices IV and V in the Supplemental Material. We have two main findings from this numerical analysis. First, the basin of attraction is always minuscule, with volumes always smaller than 001% of the total volume of the state space .Second,ξ†is almost completely nonrobust to changes in behavior in population 1. Evidence for this lies in the position of the saddle points, which have more than 998% of population 1agents choosing strategy 1, indicating that changes in the behavior of 02% of population 1agents are enough to disrupt the stability of ξ†. Thus, the exact stability analysis of the backward induction state for larger numbers of trials is undercut by a thorough numerical analysis of the dynamics in the vicinity of that state.
1366 Sandholm, Izquierdo, and Izquierdo Theoretical Economics 14 (2019) 4.2 Persistence of the stable interior rest point When agents test their strategies thoroughly, the distributions of opponents’ choices they face when testing each strategy will come to resemble the current distribution of play in the opposing population. Since agents choose the strategy whose total payoff during testing was highest, this suggests that the rest points of the resulting dynamics should approximate Nash equilibria. Indeed, when agents possess exact information, so that play adjusts according to the exact best response dynamic (Gilboa and Matsui (1991), Hofbauer (1995)), the results of Xu (2016) imply that every solution trajectory converges to the set of Nash equilibria; in centipede, all Nash equilibria entail all population 1 agents stopping immediately. While the intuition suggested above is correct for large enough numbers of trials, it is nevertheless the case that stable cooperative behavior can persist when the number of trials of each strategy is substantial. To illustrate this, we consider play in the centipede game of length d=4under the BEP(τallκβmin)dynamic. Figures 3and 4present the stable rest points of this dynamic for numbers of trials κup to 50, which we computed using numerical methods. While increasing the number of trials shifts mass toward uncooperative strategies, it is clear from the figures that this shifting takes place gradually: even with rather thorough testing, significant levels of cooperation are still maintained. We note as well that the fraction of population 2 agents who play the weakly dominated strategy [0] (always continue) becomes fixed between 7% and 65% once κ≥15,evenas the fraction of population 1 agents who play strategy [0] remains far from 0(specifically, between 28% and 18%). While surprising at first glance, these facts can be explained by considering both the expectations and the dispersions in the payoffs obtained through repeated trials of each strategy. As an illustration, consider the stable rest point when κ=32, namely ξ∗= (x∗y∗)≈((021400573802122),(063330301000657)).Letjbe a random variable that represents the payoff obtained by a population 2 agent who plays strategy jin a single random match at this state. By (2)(orFigure 1), the expected payoffs to this agent’s three strategies are E(1)=(033)·x∗=23580 E(2)=(025)·x∗=22086 E(3)=(024)·x∗=19964 From this we anticipate that the strategy weights in population 2 satisfy y∗ 1>y∗ 2>y∗ 3. To explain why these weights take the values they do, we also need to know how dispersed are the payoffs from testing each strategy. We thus compute the variances of the single-test payoffs j: Var(1)=15138 Var(2)=27223 Var(3)=17048
Theoretical Economics 14 (2019) Dynamics and cooperation in centipede 1367 Figure 3. The stable interior rest point of the BEP(τallκβmin)dynamic in the centipede game of length d=4,κ=150. Stacked bars, from the bottom to the top, represent weights on strategies [0], [1], and [2]. Using these calculations and the central limit theorem, we find that the difference between the average payoffs from 32 tests of strategy 3 and 32 tests of strategy 2 is approximately normally distributed with mean E(3)−E(2)=−02122 and standard deviation (Var (3)+Var (2))/32 ≈03720. The latter statistic is commensurate with the former. Thus the weakly dominated strategy 3 yields a higher total payoff than the dominating strategy 2 with approximate probability P(Z ≥057)≈028 and so is not a rare event. Likewise, evaluating the appropriate multivariate normal integrals shows that the probabilities of strategies 1, 2, and 3 yielding the highest total payoff are approximately 061,032,and007, figures which accord fairly well with the components of y∗. As the number of trials κbecomes larger, greater averaging reduces the variation in each strategy’s payoffs per trial. At the same time, increasing κincreases the weight x∗ 1 on stopping immediately at the expense of population 1’s other two strategies, hence reducing the differences in the expected payoffs of population 2’s strategies. This explains
1368 Sandholm, Izquierdo, and Izquierdo Theoretical Economics 14 (2019) Figure 4. The stable interior rest point in centipede of length d=4under BEP(τallκβmin) dynamics for κ=134 trials of each tested strategy. Lighter shading corresponds to larger numbers of trials. Dashed lines represent boundaries of best response regions. why the strategy weights in population 2 do not vary very much as κincreases and why the weight on the weakly dominated strategy hardly varies at all. 4.3 Convergence to cycles Figure 3 does not record rest points for certain numbers of trials above 34. For these values of κ, the population state does not converge to a rest point. Instead, our numerical analyses indicate that for all κwith empty entries in Figure 3 and all κbetween 51 and 100,theBEP(τallκβmin)dynamic converges to a periodic orbit. Figure 5 presents the cycles under the BEP(τallκβmin)dynamics for κ=50,100,and200. In all three cases, we observe substantial levels of cooperative play in population 1 over the course of the cycle, with the fraction of the population choosing to continue at the initial node varying between 050 and 083 for κ=50, between 028 and 070 for κ=100, and between 016 and 045 for κ=200. These examples illustrate that cooperative behavior can persist even when agents have substantial amounts of information about opponents’ play. From a methodological point of view, the existence of attracting limit cycles under BEP dynamics suggests that solution concepts like S(k) equilibrium and logit equilibrium that are motivated as steady states of dynamic disequilibrium processes should be applied with some caution. Existence results for such solution concepts can generally be proved by appeals to suitable fixed point theorems. But the fact that static solutions exist need not imply that any are stable, and it may happen that no static solution provides a good prediction of the behavior of the underlying dynamic process. 5. Conclusion In this paper, we introduce a class of game dynamics built on natural assumptions about the information agents obtain when revising, and we show that these dynamics lead to cooperative behavior in the centipede game. One key feature of the agents’ revision process is that conditional on the current population state, the experienced payoffs to each
Theoretical Economics 14 (2019) Dynamics and cooperation in centipede 1369 Figure 5. Stable cycles in centipede of length d=4under BEP(τallκβmin)dynamics for κ=50,100,and200. Lighter shading represents faster motion. The small circles represent the unstable interior rest points. For κ=50 and 100, shapes synchronize positions along the cycle. strategy are independent of one another. This allows cooperative strategies with suboptimal expected payoffs to be played with nonnegligible probabilities, even when the testing of each strategy involves substantial numbers of trials. The use of any such strat-
1370 Sandholm, Izquierdo, and Izquierdo Theoretical Economics 14 (2019) egy increases the expected payoffs of other cooperative strategies, creating a virtuous circle that sustains cooperative play. Appendix A: Proof of Proposition 1 A.1 Generalities Letting s=s1+s2, we denote the tangent space of the state space =X×Yby T= TX×TY ={(z1z2)∈Rs:i∈S1z1 i=0and j∈S2z2 j=0}and we denote the affine hull of by aff() =T+ξ†. Writing our dynamics as ˙ ξ=V(ξ) (D) we have V:aff() →T,andsoDV (ξ)z ∈T for all ξ∈and z∈T.Wecanthus view DV (ξ) as a linear map from Tto itself, and the behavior of the dynamics in the neighborhood of a rest point is determined by the eigenvalues and eigenvectors of this linear map. The latter are obtained by computing the eigenvalues and eigenvectors of the product matrix DV(ξ),whereV:Rs→Rsis the natural extension of Vto Rs, and is the orthogonal projection of Rsonto T, i.e., the block diagonal matrix with diagonal blocks I−1 s111∈Rs1×s1and I−1 s211∈Rs2×s2,where1=(11). Since V maps into T, the projection is only needed when there are eigenspaces of DV (ξ) that intersect both the set Tand its complement. We prove that the backward induction state ξ†is a repellor using the following argument. Computing the eigenvalues and eigenvectors of DV (ξ†)as described above, we find that ξ†is a hyperbolic rest point, meaning that all of the eigenvalues have a nonzero real part. The linearization of the dynamic (D) at rest point ξ†is the linear differential equation ˙ z=DV ξ†z(L) on T.Thestable subspace Es⊆Tof (L) is the span of the real and imaginary parts of the eigenvectors and generalized eigenvectors of DV (ξ†)corresponding to eigenvalues with a negative real part. The unstable subspace Eu⊆Tof (L) is defined analogously. The basic theory of linear differential equations implies that solutions to (L)onEsconverge to the origin at an exponential rate, that solutions to (L)onEudiverge from the origin at an exponential rate, and that the remaining solutions approach Euand then diverge from the origin at an exponential rate. Let As=Es+ξ†and Au=Eu+ξ†denote the affine spaces that are parallel to Esand Euand that pass through ξ†.InAppendix A.2,weprovethatundertheBEP(τall1βmin) dynamic, the dimensions of Esand Euare d−1and 1, and that Asis a supporting hyperplane to at ξ†. Combining these facts with fundamental results from dynamical systems theory lets us complete the proof that ξ†is a repellor. By the Hartman–Grobman theorem (Perko (2013, Section 2.8)), there is a homeomorphism hbetween a neighborhood of ξ†in aff() and a neighborhood of 0in Tthat maps solutions of (D) to solutions of (L). By
Theoretical Economics 14 (2019) Dynamics and cooperation in centipede 1377 Appendix D: Proof of Proposition 4(ii) The interior rest point ξ∗of the dynamic ˙ x=V(x)is locally stable if all eigenvalues of the derivative matrix DV (ξ∗)have negative real part. Since each entry of the derivative matrix DV(ξ∗)is a polynomial with many terms that is evaluated at a state whose components are algebraic numbers, it is not feasible to compute its eigenvalues exactly. We circumvent this problem by computing the eigenvalues of the derivative matrix at a nearby rational state ξand making use of a bound on the distances between the eigenvalues of the two matrices. This bound is established in Proposition 6. As in Appendix A,lets=s1+s2=d+2,let˙ ξ=V(ξ),V:aff() →T denote an instance of the BEP dynamics, and let V:Rs→Rsdenote the natural extension of Vto Rs. Observe that if DV(ξ) is diagonalizable, then so is DV (ξ), and all eigenvalues of the latter are eigenvalues of the former. To state the proposition, we write S=S1∪S2and omit population superscripts to define =max i∈Smax k∈S j∈S ∂2Vi ∂ξj∂ξk (11|11) (23) Proposition 6. Suppose that DV(ξ) is (complex)diagonalizable with DV(ξ) =Q× diag(λ)Q−1, and let λ∗be an eigenvalue of DV(ξ∗). Then there is an eigenvalue λiof DV(ξ) such that λ∗−λi<2 ss/2−1 trQ∗Qs/2 det(Q) k∈Sξk−ξ∗ k(24) The eigenvalue perturbation theorem (26) that begins the proof of the proposition bounds the distances between the eigenvalues of DV(ξ∗)and DV(ξ), but neither term on its right-hand side is feasible to compute. The second paragraph of the proof provides a bound on the condition number κ∞(Q) that does not require the computation of the inverse of the (algebraic-valued) eigenvector matrix Q. The third paragraph provides a bound on the norm of DV(ξ) −DV(ξ∗), which is needed because numerically evaluating of the entries of DV(ξ∗)with guaranteed precision is computationally infeasible. Two further devices that we employ to improve the bound and speed its computation are described after the proof of the proposition. Proof of Proposition 6.ForM∈Rs×s,let |||M|||∞=max 1≤i≤s s j=1|Mij |(25) denote the maximum row sum norm of M.Letκ∞(Q) = |||Q|||∞|||Q−1|||∞be the condition number of Qwith respect to norm (25). The following eigenvalue perturbation theorem (Horn and Johnson (2013, Observation 6.3.1)) follows from the Geršgorin disk theorem and the submultiplicativity of matrix norms: λ∗−λi≤κ∞(Q)DV(ξ) −DVξ∗∞(26)
1378 Sandholm, Izquierdo, and Izquierdo Theoretical Economics 14 (2019) To bound κ∞(Q),let|||M|||2denote the spectral norm of M(i.e., the largest singular value of M)andletκ2(Q) = |||Q|||2|||Q−1|||2be the condition number of Qwith respect to this norm. Since the maximum row sum and spectral norms differ by a factor of at most √s(Horn and Johnson (2013, Problem 5.6.P23)), it follows that κ∞(Q) ≤sκ2(Q) (27) Also, Guggenheimer et al. (1995)(seealsoMerikoski et al. (1997)) show that κ2(Q) < 2 det(Q)trQ∗Q ss/2 (28) To bound the final expression in (26), note that by construction, each component of the BEP dynamics ˙ x=V(x) is the difference between a sum of monomials in the components of ξwith positive coefficients and a linear term. Thus, the second derivatives of Vi(ξ) are sums of monomials with positive coefficients. Since every component of every state ξ∈is at most 1, we therefore have max ξ∈ ∂2Vi ∂ξj∂ξk (ξ)≤∂2Vi ∂ξj∂ξk (11|11) (29) Thus, the fundamental theorem of calculus, (29), and (23) imply that DV(ξ) −DVξ∗∞≤max i∈S j∈S k∈S ∂2Vi ∂ξj∂ξk (11|11)×ξk−ξ∗ k ≤ k∈Sξk−ξ∗ k(30) Combining inequalities (26), (27), (28), and (30) yields inequality (24). When applying Proposition 6, one can choose Qto be any matrix of eigenvectors of DV(ξ).Guggenheimer et al. (1995) suggest that choosing the eigenvectors to have Euclidean norm 1 (which if done exactly makes the expression in parentheses in (28) equal 1) leads to the lowest bounds. We apply this normalization in the final step of our analysis. To use this bound to establish the stability of the interior rest point ξ∗, we choose a rational point ξclose to ξ∗, compute the eigenvalues of the derivative matrix DV (ξ), and evaluate the bound from Proposition 6. The eigenvalues of DV (ξ) all have negative real part as long as ξis reasonably close to ξ∗. If ξis close enough to ξ∗that the bound is smaller than the magnitude of the real part of any eigenvalue of DV (ξ), we can conclude that the eigenvalues of DV (ξ∗)all have negative real part and, hence, that ξ∗is asymptotically stable. Selecting state ξinvolves a trade-off: choosing ξcloser to ξ∗reduces the bound, but doing so also leads the components of ξto have larger numerators and denominators, which slows the computation of the bound significantly. In all cases, we are able to choose ξsatisfactorily and to conclude that ξ∗is asymptotically stable. For further details about how the computations are implemented, see the Supplemental Material.
Theoretical Economics 14 (2019) Dynamics and cooperation in centipede 1379 Appendix E: Proof of Proposition 5 Letting K={1κ}, we can write the population 1 equations of the BEP(τallκβmin) dynamic as ˙ xi= r:S1×K→S2 ∈S1λ∈K yrλ 1 i=minargmaxk∈S1π1 k(r)!−xi where π1 k(r) = κ m=1 Akrkm (31) The result function ( λ) → rλ specifies the strategy in S2played by an agent’s match partner during the λth test of strategy for all ∈S1and λ∈K. The second piece of (31) specifies the probability of a given result, and the third piece indicates whether strategy iis the minimal optimal strategy for this result. If there are two or more occurrences of strategies from S2other than 1, then all partial derivatives of the product in (31) equal 0. Thus, for the purposes of computing the Jacobian, we need only consider results in which there are 0 or 1 match partners playing strategies other than strategy 1∈S2. These results comprise the following possibilities: (i) If all match partners play strategy 1∈S2, then strategy 1∈S1earns total payoff κ·0 and all other strategies earn total payoff κ·(−1), so strategy 1has the best experienced payoff. (ii) If the lone match against another strategy j∈S2\{1}occurs when the revising agent plays strategy 1∈S1, then total payoffs are as above and strategy 1has the best experienced payoff. (iii) If the lone match against another strategy occurs when the revising agent plays strategy i∈S1\{1}and if this match occurs against an opponent playing strategy j∈ S2\{1}, then (using the payoffs Aij defined in (2)) strategy 1is the minimal strategy earning the best experienced payoff if κ·0≥(κ −1)·(−1)+2i−2if i≤j 2j−3if i>j; otherwise, strategy iuniquely obtains the best experienced payoff. Accounting for all of these possibilities, including the fact that the matches in cases (ii) and (iii) can occur during any of the κtests of the strategy in question, we have ˙ x1=(y1)κs1+κ(y1)κs1−1s2 j=2 yj+ s1 i=2i−1 j=2 yj12j−2≤κ+ s2 j=i yj12i−1≤κ −x1+O(y−1)2 ˙ xi=κ(y1)κs1−1i−1 j=2 yj12j−3≥κ+ s2 j=i yj12i−2≥κ−xi+O(y−1)2 (32a) where y−1=s2 j=2yjand i∈S1\{1}.
1380 Sandholm, Izquierdo, and Izquierdo Theoretical Economics 14 (2019) Turning to population 2, the test results with 0 or 1 match against opponents playing strategies other than 1∈S1comprise the following possibilities: (i) If all match partners play strategy 1∈S1, then all strategies earn total payoff κ·0, so strategy 1is the minimal strategy earning the best experienced payoff. (ii) If the lone match against another strategy occurs when the revising agent plays strategy j∈S2, then strategy jearns a positive total payoff and other strategies earn total payoff 0, so strategy jhas the best experienced payoff. Accounting for both possibilities, we obtain ˙ y1=(x1)κs2+κ(x1)κs2−1 s1 i=2 xi−y1+O(x−1)2 ˙ yj=κ(x1)κs2−1 s1 i=2 xi−yj+O(x−1)2 (32b) where x−1=s1 i=2xiand j∈S1\{1}. Taking the derivative of (32a)and(32b) at state ξ†, we obtain the matrix (we write this matrix for the case of deven, so that s1=s2; roughly speaking, the case of dodd corresponds to removing the final column) DVξ†= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −10··· ··· 0κs1+ ··· ··· + 0−1 0κ12i−2≥κ··· ··· κ12i−2≥κ κ12j−3≥κ 0 0··· ··· 0−10κ12j−3≥κ··· κ12j−3≥κκ12i−2≥κ κs2κ··· ··· κ−10··· ··· 0 0κ··· ··· κ0−1 0 0 0κ··· ··· κ0··· 00 −1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (33) Each +above represents the number that makes the column sum in the block equal κs1. If all indicator functions in the upper-right block of DV(ξ†)equal 0, then each +in (33)equalsκs1, implying that this block is the zero operator on TY.Inthiscase,(33) acts as a block triangular matrix on T, and so its lone eigenvalue with respect to Tis −1, implying that ξ†is stable. To check that the indicators are all 0 when dis even, it is enough to consider the indicator for entry j=i=s2≡1 2d+1, which is 0 if and only if κ≥d+1.Whendis odd, it is enough to check the indicator for entry j=i=s1−1≡1 2(d +1), which is 0 if and only if κ≥d. We conclude that ξ†is asymptotically stable in these cases.
Theoretical Economics 14 (2019) Dynamics and cooperation in centipede 1381 To show that ξ†is unstable in the remaining cases (when κ≥2and d≥3), write χκ ij =12i−2≥κif i≤j 12j−3≥κif i>jχ κ i = s2 j=2 χκ ij and χκ = s1 i=2 s2 j=2 χκ ij for i j ≥2. A straightforward calculation shows that λ=κ"χκ −1is an eigenvalue of DV(ξ†)that corresponds to eigenvector z=#−χκ χκ 2χκ s1|−s2−1"χκ "χκ "χκ $ Since κ≥2,λis positive whenever at least one of the indicators in DV(ξ†)equals 1. Combining this with the previous argument, we conclude that ξ†is unstable whenever it is not asymptotically stable. References Akritas, A. G. (2010), “Vincent’s theorem of 1836: Overview and future research.” Journal of Mathematical Sciences, 168, 309–325. [1350,1375] Basu, Kaushik (1994), “The traveler’s dilemma: Paradoxes of rationality in game theory.” American Economic Review Papers and Proceedings, 84, 391–395. [1351] Becker, Eberhard, Maria Grazia Marinari, Teo Mora, and Carlo Traverso (1994), “The shape of the Shape Lemma.” In ISSAC ’94: Proceedings of the International Symposium on Symbolic and Algebraic Computation (J. von zur Gathen and M. Giesbrecht, eds.), 129–133, ACM. [1374] Ben-Porath, Elchanan (1997), “Rationality, Nash equilibrium and backwards induction in perfect-information games.” Review of Economic Studies, 64, 23–46. [1348] Benaïm, Michel and Jörgen W. Weibull (2003), “Deterministic approximation of stochastic evolution in games.” Econometrica, 71, 873–903. [1348,1353] Berkemer, Rainer (2008), “Disputable advantage of experience in the travelers’ dilemma.” Unpublished paper, Technical University of Denmark, Deptartment of Mathematics. [1351] Binmore, Ken (1987), “Modeling rational players: Part I.” Economics and Philosophy,3, 179–214. [1348] Binmore, Ken (1998), Game Theory and the Social Contract, Volume 2: Just Playing.MIT Press, Cambridge. [1352] Björnerstedt, Jonas and Jörgen W. Weibull (1996), “Nash equilibrium and evolution by imitation.” In The Rational Foundation of Economic Behavior (Kenneth J. Arrow, Enrico Colombatto, Mark Perlman, and Christian Schmidt, eds.), 155–171, London: MacMillan. [1352]
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