Strategic complementarities in a model of commercial media bias

Kerkhof, Anna; Münster, Johannes Article Strategic complementarities in a model of commercial media bias Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Kerkhof, Anna; Münster, Johannes (2025) : Strategic complementarities in a model of commercial media bias, Games, ISSN 2073-4336, MDPI, Basel, Vol. 16, Iss. 3, pp. 1-46, https://doi.org/10.3390/g16030021 This Version is available at: https://hdl.handle.net/10419/330135 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 Editors: Fabrizio Germano and Ulrich Berger Received: 25 October 2023 Revised: 14 December 2024 Accepted: 19 December 2024 Published: 23 April 2025 Citation: Kerkhof, A., & Münster, J. (2025). Strategic Complementarities in a Model of Commercial Media Bias. Games,16(3), 21. https://doi.org/ 10.3390/g16030021 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/). Article Strategic Complementarities in a Model of Commercial Media Bias Anna Kerkhof 1and Johannes Münster 2,* 1Ifo Institute for Economic Research, and CESifo, University of Munich, 80539 Munich, Germany; [email protected] 2Department of Economics, University of Cologne, 50923 Cologne, Germany *Correspondence: [email protected]; Tel.: +49-221-470-4411 Abstract: Media content is an important privately supplied public good. While it has been shown that contributions to a public good crowd out other contributions in many cases, the issue has not been thoroughly studied for media markets yet. We show that in a standard model of commercial media bias, qualities of media content are strategic complements, whereby investments into quality can crowd in further investments and engage competitors in a race to the top. Therefore, financially strong public service media can mitigate commercial media bias: the content of commercial media can be more in line with the preferences of the audience and less advertiser-friendly in a dual (mixed public and commercial) media system than in a purely commercial media market. Keywords: commercial media bias; public service media; advertising; two-sided markets; supermodular games; strategic complements; public goods JEL Classification: C70; H41; L13; L51; L82 1. Introduction Media content belongs to the most important cases of privately supplied public goods. Its consumption is non-rival, and in many cases like free TV or freely available Internet content no exclusion is taking place. Media content differs markedly from other public goods, though, because media outlets typically rely on advertising revenues instead of charging their consumers a pecuniary price. The economic analysis of the private supply of media content as a public good must take this multi-sided nature of media markets into account (Anderson & Coate,2005). Recent literature has made major progress in this research area (see Anderson & Jullien,2015;Jullien et al.,2021). One important result from the theory of private public good supply is that, under fairly general conditions, private contributions to a public good are strategic substitutes, i.e., higher private contributions to the public good are crowding out other private contributions (for overviews, see Chapter 6 of (Batina & Ihori,2005), and Finding F9 in (Buchholz & Sandler,2021)). Surprisingly, this issue has not been thoroughly examined for media markets, even though it is highly relevant for the welfare analysis of media policy. E.g., in discussions about the proper role and scope of public service media (PSM), one crucial question is whether raising the quality of a regulated PSM will increase or decrease the quality of its commercial—i.e., profit-maximizing—competitors. There are two conflicting views. On the one hand, PSM could crowd out private investment and innovation in media markets. E.g., the existence of PSM may lead to less entry of Games 2025,16, 21 https://doi.org/10.3390/g16030021 Games 2025,16, 21 2 of 46 commercial media; see (Berry & Waldfogel,1999) for empirical evidence. Similarly, (Armstrong & Weeds,2007a) show that in a duopoly where a PSM and a commercial broadcaster compete, raising the quality of PSM partially crowds out the commercial broadcaster and lowers its quality. This reasoning is echoed by regulation authorities like Ofcom (Ofcom, 2004) in the UK and the Scientific Advisory Board at the Federal Ministry of Finance in Germany (Wissenschaftlicher Beirat beim Bundesministerium der Finanzen,2014). However, PSM might also foster a “competition for quality”, whereby public and commercial media compete for audiences. This reasoning goes back to (Coase,1947), pondering that PSM might induce a “natural rivalry to furnish the most attractive programs” (p. 197). Indeed, recent empirical evidence suggests that in countries where PSM invest into high-quality media content, the quality of commercial media tends to be high, too (Simon, 2013). Similarly, (Sehl et al.,2020) find that, controlling for GDP, per capita revenues of PSM and commercial broadcasters are positively correlated across EU countries. These correlations are in line with a crowding in effect of PSM, i.e., the presence of strong PSM coincides with flourishing commercial media.1 In this paper, we show that in a model of commercial media bias, program qualities in terms of unbiased reporting are strategic complements rather than strategic substitutes. 2 Unbiased reporting here refers to a program that fully and truthfully reports facts as opposed to withholding information. E.g., advertisers might prefer the media to hide unfavorable facts about their products; prime examples include the tobacco and carbonemitting industries. Viewers prefer high program quality, while advertisers prefer the opposite. The strategic complementarity stems from the media’s fundamental trade-off in these models: Raising program quality increases the value of the program for the audience but decreases the willingness to pay of the advertisers to reach consumers. 3 The latter effect becomes less important when a media company has a smaller audience; hence, its incentives to raise program quality are higher. Thus, in a media market with both PSM and commercial media, raising the PSMs’ program quality reduces the commercial media’s audiences and thereby also their implicit cost of increasing their own program quality. As a result, the PSM crowd in program quality and engage the commercial media in a race to the top. Our main model focuses on PSM and commercial media whose programs are freely available to the consumers. However, our results generalize to a model featuring both freely available and pay media, when program quality involves revealing information that the media already possess. We also discuss conditions under which our findings generalize to multidimensional strategy spaces, spillover effects of program quality on advertising revenue of other media outlets, endogenous entry and exit, and biases of PSM.4 Our paper relates to four strands of literature. First, we contribute to the literature on commercial media bias. Several empirical papers document the effect of advertising on media coverage in terms of mutual fund recommendations (Reuter & Zitzewitz,2006), product mentions (Gambro & Puglisi,2015), coverage of government scandals (Tella & Franceschelli,2011) and climate change (Beattie,2020). We present a fairly standard model of commercial media bias. Our model is in many ways similar to the models studied by (Ellman & Germano,2009), (Germano & Meier,2013), and (Kerkhof & Münster,2015), as it captures bias through a program that caters to the preferences of advertisers rather than consumers. Our paper is especially close to (Ellman & Germano,2009) and (Germano & Meier,2013) who show that competition in media markets mitigates commercial bias, and to (Kerkhof & Münster,2015) who find that competition between media outlets increases the likelihood that a cap on advertising quantities is welfare enhancing. Relatedly, (Blasco et al.,2016) find that if the media can raise their audience share through reducing their bias, then competition in the market may also increase the expected program quality. 5 Games 2025,16, 21 3 of 46 These predictions are in line with the empirical results of (Beattie et al.,2021) who find that newspapers provide less coverage of car recalls by their advertisers, but competition for readers mitigates this bias. Similarly, (Focke et al.,2016) show that commercial media bias is likely mitigated by reputational concerns on behalf of the media, e.g., if they face a demanding audience. In contrast to the existing literature, the present study considers competition between commercial media and PSM, where PSM are not profit-maximizing, potentially regulated, and do not depend on advertising revenue to fund their operations. This allows us to inform policy debates regarding the proper role and scope of PSM in media markets. Moreover, in contrast to previous work, the present paper explicitly models competition between media outlets as a supermodular game, enabling us to draw fairly general conclusions regarding the impact of raising PSMs’ budget on the program quality provided by commercial media. Second, we advance the broad research on the private supply of public goods (Batina & Ihori,2005;Bergstrom et al.,1986). The provision of public goods via advertising is studied by (Anderson & Coate,2005;Luski & Wettstein,1994). These papers do not study media bias, however. Third, our paper relates to the literature on supermodular games, i.e., games in which the best response of any player is increasing in the actions of its competitors (Frankel et al.,2003;Milgrom & Roberts,1990;Topkis,1979;Van Zandt & Vives,2007;Vives,1985, 1990,2005a,2005b). To the best of our knowledge, we are the first to apply the theory of supermodular games to a model of commercial media bias. This approach allows us to obtain fairly general results in a model with many asymmetric media outlets. Specifically, we show that in our model of commercial media bias, program qualities are strategic complements rather than strategic substitutes. Fourth, as a consequence of strategic complementarities, public investments into program quality induce commercial media to provide high quality, too. Hence, our results support media policies that advocate financially strong PSM. In this way, we also contribute to the economics literature on PSM (see (Armstrong & Weeds,2007b;Strömberg,2015; Weeds,2020) for surveys). To the best of our knowledge, the issue how PSM affect the program of commercial media has not been studied yet in the literature on commercial media bias. Other aspects of this debate have, of course, been analyzed; in addition to the empirical literature referenced above, several theoretical studies on the market impact of PSM exist. (Armstrong & Weeds,2007a) study investments in a vertical quality dimension. (Richardson,2006) investigates how a publicly-provided radio station offering local programs affects the provision of local programs by commercial stations. (Garcia Pires,2016) compares program diversity in commercial versus mixed public and private duopolies. Our paper complements this line of research by studying commercial media bias. E.g., neither (Armstrong & Weeds,2007a) nor (Richardson,2006) consider advertisers who value program qualities in terms of (un-)biased reporting. In (Armstrong & Weeds, 2007a), viewers are ad averse and higher advertising quantities reduce viewers’ utility. PSM maximize viewer welfare, whereby viewers are better off than in a purely commercial market. However, in contrast to our paper, this is because PSM partially crowd out commercial media, whereby subscription prices and advertising quantities decrease. Similarly, (Richardson,2006) shows that in a Hotelling model with ad averse viewers, PSM reduce profits of commercial media, but increase viewer welfare. Thus, in both models, viewers are better off because audience-friendly PSM displace commercial media. Our paper considers a different mechanism: financially strong PSM enhance viewers’ utility because they crowd in program qualities by commercial media. The remainder of this paper is structured as follows. Section 2introduces our theoretical framework. In Section 3, we demonstrate that program qualities in terms of Games 2025,16, 21 4 of 46 unbiased reporting are strategic complements, which is our main finding, and describe the implications for crowding in effects of PSM. Section 4considers the case where some commercial media are pay media. Section 5discusses several extensions of our model. Section 6concludes. 2. Model This section introduces a fairly standard model of commercial media bias (Ellman & Germano,2009;Germano & Meier,2013;Kerkhof & Münster,2015), see (Blasco et al.,2012) for a survey). Consider a model with n commercial media denoted by 1,..., n and m PSM denoted n+ 1, . . . , n+m . The set of commercial media is denoted by C={1, . . . , n} , the set of PSM is P={n+1, . . . , n+m} . Each media outlet i∈C∪P chooses a program quality vi∈Vi⊂R+ . (An extension to multidimensional strategy spaces is considered in Section 5). Program quality vi is about unbiased reporting, i.e., about fully and truthfully reporting facts, as opposed to withholding information. 6 The audience prefers high program quality, while advertisers prefer the opposite. We assume that the strategy sets Vi are compact and contain vi=0. A consumer’s utility from consuming outlet i is ui=fi(vi) , where fi is continuous, strictly increasing, and satisfies fi(0)= 0. Unless otherwise noted, we simply assume fi(vi)=vi . In Sections 2and 3, nothing is lost in setting ui=vi ; the distinction between utility ui and program quality vi becomes important when considering pay media or multidimensional strategies. For a commercial outlet i∈C , let uC −i=(u1, . . . , ui−1,ui+1, . . . , un) denote the vector of the utilities of i ’s commercial competitors, uP=(un+1, . . . , un+m) the vector of utilities of the PSM, and u−i=uC −i,uP.7 The size of the audience of a media outlet is denoted by si . We impose the following assumptions.8 Assumption 1. For all i∈C , si is positive, continuous, weakly increasing in ui , and weakly decreasing in ujfor all j ∈P∪C\{i}. Assumption 1 is reasonable if consumers care about quality, and the media outlets are substitutes for the consumers. Assumption 2. For all i ∈C,sihas weakly increasing differences in (ui,u−i). If si is twice continuously differentiable, and the strategy spaces are intervals, Assumption 2 means that ∂2 ∂uj∂uisi≥0 for all j=i . 9 Note that Assumption 2 only assumes that the differences are weakly increasing. In particular, it is fulfilled in the case of constant differences, where the above inequality holds with equality. As we discuss in detail in Appendix A, Assumptions 1 and 2 are satisfied by many— but not all—models for audience demand that are frequently used in media economics. For example, any model where si is linear in ui and uj for all j=i , and does not include any interaction terms, satisfies Assumption 2 because si has constant differences in (ui , u−i) . This class of models comprises the Hotelling duopoly model of horizontally differentiated goods enriched by a vertical quality differentiation, and generalizations of the Hotelling model to more than two outlets such as the Spokes model and the Salop circle model. Similarly, si has constant differences in (ui , u−i) in representative consumer models with quadratic utility functions. See Appendix Afor the functional forms of si in these models, and for references to publications in media economics using these specifications. Games 2025,16, 21 5 of 46 Note, however, that our assumptions are far more general than simply assuming linear demand functions. For example, when the audience demand functions include linear and quadratic terms, si(ui,u−i)=ai+biui−∑ j=i bijuj+∑ k cikukui, Assumption 2 holds as long as cik ≥0 for all i∈Cand all k=i. On the other hand, the logit model violates Assumption 2 whenever there are n≥ 2 commercial outlets. We deal with possible violations of Assumption 2 in two distinct ways. First, in Appendix Fwe give a sufficient condition for our main results to hold when Assumption 2 is violated, and thereby show the robustness of our results to sufficiently small violations of Assumption 2. Second, we explore our model with a weaker version of Assumption 2: Assumption (2 log) For all i∈C , ln(si) has weakly increasing differences in (ui,u−i) . Given Assumption 1, Assumption 2 implies Assumption (2 log), but not vice versa. For example, the logit model violates Assumption 2, but satisfies assumption (2 log) (see Appendix F). Moreover, the nested logit model—which is often used in empirical studies of audience demand in media economics (see Berry & Waldfogel,2015 for a survey)—satisfies Assumption (2 log) as well (see Appendix F). 10 For some of our results, Assumption (2 log) is sufficient. Unless otherwise noted, below we will assume that Assumption 2 holds; we will explicitly state when we weaken it to Assumption (2 log). Denote the advertising revenue of outlet i , per member of the audience, by Ri . A crucial assumption in models of advertiser bias is that, for a given audience, ad revenue depends negatively on program quality: Assumption 3. For all i∈C , Ri is positive, continuous, weakly decreasing in vi ,and independent of vjfor all j =i. By Assumption 3, Ri is independent from the program quality of other outlets as in (Ellman & Germano,2009) and (Kerkhof & Münster,2015). (Germano & Meier,2013) model spillover effects of program quality on the advertising revenue of other outlets; we will discuss spillover effects in Section 5.11 Each media outlet ihas a cost ci(vi)that may depend on its program quality.12 Assumption 4. For all media outlets i∈C∪P , ci is continuous, weakly increasing in vi ,and zero at vi=0. We distinguish between two cases. First, program quality could be about fully and truthfully reporting facts that the media already have. In this case, the only cost of quality is lower advertising revenue, but there is no additional direct cost of obtaining the information in the first case. Formally, in the current model it means that ci(vi)= 0 for all vi∈Vi . We refer to this case as “withholding facts”. Second, program quality could also be about investigative journalism, about establishing new facts and information. Then it seems plausible that ci(vi) is strictly increasing in vi . For example, the media might have to hire more journalists to increase program quality (see (Hamilton,2016) for a detailed description of the economics of investigative journalism). We refer to this case as “investigating facts”. Arguably, the case of withholding facts is highly relevant for the study of commercial media bias; indeed several papers in the literature focus on this case (Blasco et al.,2016; Ellman & Germano,2009;Germano & Meier,2013;Kerkhof & Münster,2015). For instance, advertising has been known to influence editors on crucial topics such as the health risks of smoking (e.g., (Bagdikian,2008) Chapter 12) and climate change (Beattie,2020;Boykoff & Boykoff,2004). The scientific facts about these topics were long well established and easily Games 2025,16, 21 6 of 46 accessible, but media coverage and public perception significantly lagged in time behind the scientific consensus. Moreover, as shown in (Beattie,2024), commercial media bias in the tone of coverage about climate change, measured based on comparisons of environmental and skeptical texts, can have important behavioral consequences, and merely changing the tone of coverage does not impact its cost. On the other hand, pressure from advertisers may also deter media from investigating facts. Our main results on freely available media do not depend on whether we study withholding or investigating facts. For pay media, we show that the distinction matters. Commercial media in our main model are funded by advertising, and their program is freely available for consumers; pay media will be considered in Section 4. The profit of a commercial media outlet i=1, ..., nis (substituting vi=uiand v−i=u−iinto si) πi(vi,v−i)=si(vi,v−i)Ri(vi)−ci(vi). Commercial outlet i maximizes πi(vi,v−i) by choosing vi∈Vi . Note that we disregard fixed costs which could be saved by going out of business; we defer a discussion of exit and entry to Section 5. The PSM in our model are not-for-profit and financed independent of advertising. Their program is freely available to all consumers. 13 The budget of PSM i∈P is bi . We assume the PSM spend their budget to maximize consumer utility by choosing vi∈Vi subject to ci(vi)≤bi . 14 The feasible sets Vi⊂R+ are compact, contain vi= 0, and may depend on the budget bi . We assume that a larger budget enlarges the feasible set: if bi<b′ i , then Vi(bi)⊆Vib′ i . The model allows for inefficiencies of PSM, since different media outlets can have different cost functions and different feasible sets of program quality. We discuss potential biases of PSM in Section 5. Some (but not all) of our considerations below impose the additional assumption that a sufficiently high program quality is necessary for a PSM to attract an audience. To express this formally, for i∈P let vP −i=(vn+1, ..., vi−1,vi+1,..., vn+m) denote the vector of program qualities of the other PSM. Assumption 5. A PSM outlet with zero program quality (vi=0) attracts no audience: si0, vC,vP −i= 0for all i∈P and vC,vP −i , and demand for the other media outlets is as if outlet i did not exist. Assumption 5 seems reasonable when the audience has a sufficiently attractive outside option not to consume any media. Note that under Assumption 5, a PSM with an insufficient budget cannot produce a program that attracts any audience; then the game reduces to a game between the remaining media outlets only. We will explicitly indicate where we use Assumption 5. 3. Main Results Consider the PSM first. Lemma 1. A PSM i chooses vi=¯ vi(bi):=max vi∈Vi(bi){vi|ci(vi)≤bi}. Moreover, ¯ vi(bi) is weakly increasing in bi ,and independent of the strategies of the other media outlets. Games 2025,16, 21 7 of 46 Proof. Outlet i∈Psolves max vi∈Vi vis.t. ci(vi)≤bi. An increase of bi relaxes the PSM’s budget constraint and enlarges the feasible set Vi , hence ¯ vi is weakly increasing in bi . Moreover ¯ vi is unique and independent of the strategies chosen by the other media outlets. We now turn to the commercial media. Lemma 1 allows us to view the game between the commercial media as parameterized by the budgets of the PSM b:=(bn+1, ..., bn+m) . Let ¯ vP(b)=(¯ vi(bi))m i=n+1 denote the vector of program qualities chosen by the PSMs. For i∈C, let ˜ πivi,vC −i,b:=πivi,vC −i,¯ vP(b) and let Γb=C,(˜ πi)n i=1,Πn i=1Vi denote the resulting game between the commercial media outlets: the set of players is C, payoff functions are ˜ πi, and strategy spaces are Vi. To state our main result, we need the concept of a parameterized supermodular game. 15 Consider a family of games with set of players N , strategy spaces Xi , and payoff functions ui parameterized by t in some partially ordered set of parameters values T . The game (N,(ui,(Xi)i∈N)i∈N,T) is a parameterized supermodular game if, for each i∈N: (i) Xi⊆Rmi is a lattice and is compact, (ii) ui(xi,x−i,t) is continuous in xi for fixed x−i and t , (iii) ui(xi,x−i,t) is supermodular in xi and has weakly increasing differences in (xi;x−i,t) . Proposition 1. Γbis a parameterized supermodular game. Proof. The strategy spaces Vi⊂R+ are compact by assumption, hence compact lattices, and the objective functions ˜ πiare continuous in vifor fixed v−iand b. Next, we show that πi has weakly increasing differences in (vi,v−i) . For simplicity of the exposition, we will assume here that the functions Ri , si and ci are differentiable; Appendix Bgives the proof without assuming differentiability. From πi(vi,v−i)=si(vi,v−i)Ri(vi)−ci(vi) we obtain ∂πi ∂vi=∂si(vi,v−i) ∂viRi(vi)+si(vi,v−i)R′ i(vi)−c′ i(vi) and ∂2πi ∂vj∂vi=∂2si(vi,v−i) ∂vj∂viRi(vi)+∂si(vi,v−i) ∂vjR′ i(vi)≥0, j=i(1) where the inequality follows because of ∂2si(vi,v−i) ∂vj∂vi≥ 0 by Assumption 2, ∂si(vi,v−i) ∂vj≤ 0 by Assumption 1, and R′ i(vi)≤0 by Assumption 3. We have shown that πi has weakly increasing differences in (vi,v−i) . Therefore, ˜ πi has weakly increasing differences in vi,vC −i . Moreover, πi has weakly increasing differences in vi,vP . It remains to show that ˜ πi has weakly increasing differences in (vi,b) . By Lemma 1, ¯ vk(bk) is increasing in bk , while ¯ vk′ does not depend on bk for k′=k . Since πi has weakly increasing differences in vi,vP , it follows that ˜ πi has weakly increasing differences in (vi,b). Proposition 1 shows that the program qualities are strategic complements. The economics of the result is straightforward. The fundamental trade-off for a commercial outlet in a model of commercial media bias is as follows: providing a program in line with the preferences of the audience attracts a bigger audience, but leads to lower advertising revenue per consumer. If the program qualities of competing media increase, the audience of Games 2025,16, 21 8 of 46 a given outlet is smaller, hence also the implicit cost of increasing its own program quality. The logic is closely related to the finding in (Germano & Meier,2013) that withholding facts typically increases with the concentration of ownership on the media market, which has found empirical support in (Beattie et al.,2021). Note that violations of Assumption 2 do not necessarily overturn Proposition 1. In Appendix F, we give a sufficient condition for Proposition 1 to hold if Assumption 2 is violated. Moreover, a different sufficient condition is available in the case of hiding information: Corollary 1. Suppose that Assumptions 1, (2 log), and 3 hold. Consider the case of hiding information, i.e., ci(vi)= 0for all vi∈Vi . Suppose that si and Ri are strictly positive for all (vi,v−i)∈Vi×V−i. Then Γbis a parameterized supermodular game. Proof. We only give the proof assuming differentiability for simplicity. Commercial outlet i maximizes πi(vi , v−i) = si(vi , v−i)Ri(vi) . By assumption, si(vi , v−i)Ri(vi)> 0. Thus the profit maximizing quality of firm ialso has to maximize ln πi=ln si+ln Ri. Since ln si has weakly increasing differences by Assumption (2 log), and Ri is independent of vj for j=i , it follows that ln πi has weakly increasing differences in (vi,v−i) . The rest of the proof is as in the proof of Proposition 1. The logit model, and the nested logit model, are two frequently used audience demand functions in media economics. As mentioned above, they satisfy Assumption (2 log). Corollary 1 thus implies that in the case of hiding information our main results apply to the (nested) logit model. Leveraging the theory of supermodular games (see (Vives,2005a), (Vives,2005b) or (Sarver,2023) for expositions) allows us to generate fairly general results in our model featuring many asymmetric media outlets. Denote a strategy profile in game Γb by vC= (v1, ..., vn) . The following corollary collects standard results for parametrized supermodular games that are useful in our context: Corollary 2. If Γb is a parameterized supermodular game, then Γb has, for any b ,a lowest equilibrium vC,low and a highest equilibrium vC,high ,such that any equilibrium vC satisfies vC,low ≤vC≤vC,high .Moreover, the equilibria vC,low and vC,high are weakly monotone increasing in b. In particular, an equilibrium exists; the standard proof for equilibrium existence in supermodular games uses the Tarski fixed point theorem. Turning to the comparative statics, note Corollary 2 does not imply that all equilibria are monotone increasing in b : only the highest and lowest equilibria are guaranteed to be weakly monotone increasing in b (see (Sarver,2023) for a general elaboration of this point for supermodular games). 16 When the equilibrium is unique, a stronger monotone comparative static result is available, which we highlight in the following Proposition 2. Proposition 2. Suppose Γb is a parameterized supermodular game and has a unique equilibrium. Then the equilibrium program quality of each commercial media outlet i∈C is weakly increasing in the budget bjof any PSM j ∈P. Proposition 2 states that PSM do not crowd out, and may even crowd in program quality, in line with the idea that PSM engage commercial media in a race to the top. To Games 2025,16, 21 15 of 46 6. Conclusions In this paper, we show that in a model of commercial media bias, program qualities in terms of unbiased reporting are strategic complements rather than strategic substitutes. The strategic complementarity stems from the media’s fundamental trade-off in these models: Increasing program quality increases the value of the program for the audience but decreases the willingness to pay of the advertisers to reach consumers. The latter effect becomes less important when a media company has a smaller audience; hence, its incentives to increase program qualities are higher. Thus, in a media market with both PSM and commercial media, PSMs’ with high program qualities give commercial media incentives to provide high quality themselves, too. As a result, the PSM crowd in quality and engage the commercial media in a race to the top. This is in line with recent empirical evidence on public and private investments into program quality (Sehl et al.,2020;Simon,2013). Our results hold under fairly general conditions. One important assumption is that audience demand functions have weakly increasing differences, a condition met by various standard demand functions. These include linear demand functions, quadratic demand functions with positive coefficients for the interaction terms, as well as models such as Hotelling, Salop, and Spokes. We also give sufficient conditions for our results to hold for audience demand functions that do not have weakly increasing differences, such as the logit model. While our main analysis considers media outlets who offer their programs for free, our results also extend to pay media when commercial media bias is about withholding facts that the media already have. Similarly, our results hold for multidimensional strategy spaces in the case of withholding facts; for the case of investigating facts, we provide conditions under which our main results generalize. Arguably, advertising revenue of all outlets might be negatively affected when some outlets report about deficiencies of a product, hence there may be spillover effects. However, we show that our results are robust when these spillover effects are small compared to the direct effect of an outlet’s own program quality on its advertising revenue. If entry or exit of commercial media was possible, commercial entrants would have to provide sufficiently high quality in order to overturn our main results. Finally, we point out that our results can allow for some biases in PSM as long as a higher budget of a PSM will translate into less severe biases of this outlet. We have also show, however, that program qualities are not always strategic complements. We give examples that show that, for pay media (or multidimensional strategy spaces) in the case of costly investigative journalism, higher quality PSMs can increase or decrease the quality of commercial media outlets, depending on details of the model. Moreover, strong PSM may prevent entry of commercial outlets in the first place. Furthermore, higher budgets for PSMs could, in case of politically captured PSMs, increase their biases, and by strategic complementarities the biases of commercial media would increase as a consequence. The paper contributes to recurrent media policy debates about the proper role and scope of PSM. While several regulation authorities fear that raising the program quality of PSM could crowd out private investments into program quality, our results support policies that advocate strong and financially well-equipped PSM. Reductions in the funding of PSM might result in a worse media landscape altogether. Our insights might be especially important for modern media markets like social media, where systematic quality controls are missing and quality standards are often claimed to be low (Zhuravskaya et al.,2020). While PSM have typically played a minor role here, our results encourage PSM to develop a stronger presence and provide high-quality content on social media, too. This reasoning is in line with recent scholarly advances calling on PSM to become “Public Service Internet platforms” with the objective to provide Games 2025,16, 21 16 of 46 opportunities for public debate, participation, and the advancement of social cohesion (Unterberger & Fuchs,2021). An interesting avenue for future research would be to test our predictions empirically. In particular, we hypothesize that an increase in PSMs’ budget would ceteris paribus translate both into higher program qualities of the PSM themselves as well as into higher program qualities of the PSMs’ commercial competitors. Similarly, reductions in PSM budgets would lead to lower program qualities of both PSM and commercial media. However, the identification of a causal relationship between PSM budgets and program qualities would require some exogenous variation in PSM budgets. Author Contributions: Conceptualization, A.K. and J.M.; formal analysis, J.M.; writing—original draft preparation, A.K. and J.M.; writing—review and editing, A.K. and J.M. All authors have read and agreed to the published version of the manuscript. Funding: This project is funded by the Bavarian State Ministry of Science and the Arts in the framework of the bidt Graduate Center for Postdocs. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC2126/1-390838866. Data Availability Statement: The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author. Acknowledgments: We thank Lara Mai for excellent research assistance. Conflicts of Interest: The authors declare no conflict of interest. Appendix A. Assumptions 1 and 2 Are Satisfied by Many Frequently Used Models of Audience Size in Media Economics In the media economics literature, there are four commonly used ways to model audience size when media quality matters (see the surveys by (Anderson & Gabszewicz, 2006;Anderson & Jullien,2015). These models are also widely used in other fields, see (Huang et al.,2013) for a survey. This appendix discusses which of these models satisfy Assumptions 1 and 2, and gives references to publications in media economics that employ these models. First, Hotelling model of horizontally differentiated goods, enriched by a vertical quality differentiation. In these models, there are two competing outlets. When all market shares are positive, siui,uj=1 2+ui−uj 2τ, where τ> 0 is a parameter for the degree of product differentiation. This specification is used (with ui=vi−pi where pi is the price) for example, in (Armstrong & Weeds,2007a, 2007b;Bisceglia,2023;D’Annunzio,2017;Gonzalez-Maestre & Martínez-Sánchez,2015; Li et al.,2023;Li & Zhang,2016;Liu et al.,2004;Stennek,2014). Generalizations to more than two competing outlets include the Spokes model (Chen & Riordan,2007) used by (Germano & Meier,2013), and the frequently used Salop circle model (used by (Kerkhof & Münster,2015) to study commercial media bias 24 ). Assuming all market shares to be strictly positive, these models satisfy Assumptions 1 and 2; in particular, si has constant differences in (ui,u−i). Second, representative consumer models with a quadratic utility function are used in media economics by several papers, including (Dewenter et al.,2011;Godes et al., 2009;Kind et al.,2007,2009,2016;Motta & Polo,2003). The model can allow for firm specific demand intercepts (see (Choné & Linnemer,2020)) and in this way be used to study Games 2025,16, 21 17 of 46 competition in qualities (as in (Banker et al.,1998) and (Motta & Polo,2003)). The resulting audience demand is si(ui,u−i)=ai+biui−∑ j=i bijuj, where ai , bi> 0 and bij ≥ 0. In this class of models, Assumptions 1 and 2 hold; in particular, sihas constant differences in (ui,u−i). A third type of model of audience size is the random utility model of discrete choice. The utility of choosing outlet i is composed of ui+εi , the εi are assumed to be i.i.d. distributed, and the consumer chooses the outlet offering the highest utility. Assuming the εito be Gumbel (or extreme value type I) distributed, this results in the logit model si=exp(µui) ∑n+m j=0expµuj, where j= 0 corresponds to the outside option, and µ> 0 is a parameter related to the variance of the εi . The logit model violates Assumption 2 whenever there are n≥ 2 commercial outlets, but it satisfies Assumption (log 2). In addition to the logit model, the nested logit model—which is often used in empirical studies of audience demand in media economics (see (Berry & Waldfogel,2015) for a survey)—satisfies Assumption (2 log) as well (see Appendix F). The log-separable model ((Bernstein & Federgruen,2004), see also (Huang et al.,2013)) also satisfies Assumption (2 log) when all prices are zero. Fourth, models of vertical quality differentiation in the tradition of (Mussa & Rosen, 1978); see for example (Roger,2017) for an application to media economics. These models have the property that, if all goods are available for free, then all consumers choose the good with the highest quality. Therefore, the resulting audience functions for freely available media are discontinuous in quality at the highest quality of the competitors, violating our assumptions. (With positive prices, the discontinuities are smoothed out because consumers differ in their willingness to pay for quality.) While these are valuable models for pay media, we do not consider them in this paper, where our main focus is on free media. Appendix B. Weakly Increasing Differences Without Differentiability In the proof of Proposition 1, we assumed that the functions si , Ri , and ci are differentiable in order to prove that πi has weakly increasing differences in (vi,v−i) . In this appendix we give the proof without assuming differentiability. Consider one outlet j=i , hold all other vk(k=i,j) constant and suppress them in the formulas to avoid notational clutter. Then πivi,vj=sivi,vjRi(vi)−ci(vi). Suppose that vh i>vl iand vh j>vl j. Then πivi,vh j−πivi,vl j=sivi,vh j−sivi,vl jRi(vi) and πivh i,vh j−πivh i,vl j−πivl i,vh j−πivl i,vl j =sivh i,vh j−sivh i,vl jRivh i−sivl i,vh j−sivl i,vl jRivl i =sivh i,vh j−sivh i,vl j−sivl i,vh j−sivl i,vl jRivh i +sivl i,vh j−sivl i,vl jRivh i−Rivl i Games 2025,16, 21 18 of 46 By Assumption 2, sihas weakly increasing differences in vi,vj, i.e., sivh i,vh j−sivl i,vh j≥sivh i,vl j−sivl i,vl j or equivalently sivh i,vh j−sivh i,vl j≥sivl i,vh j−sivl i,vl j. Since Rivh i≥0, it follows that sivh i,vh j−sivh i,vl j−sivl i,vh j−sivl i,vl jRivh i≥0. Moreover, by (2) sivl i,vh j≤sivl i,vl jand by (3), Rivh i≤Rivl i, thus sivl i,vh j−sivl i,vl jRivh i−Rivl i≥0. It follows that πivh i,vh j−πivh i,vl j≥πivl i,vh j−πivl i,vl j or equivalently πivh i,vh j−πivl i,vh j≥πivh i,vl j−πivl i,vl j i.e., πihas weakly increasing differences in vh i,vh j. Appendix C. Pay Media and Investigative Reporting: An Example In this appendix we consider an example of a pay media outlet in the case of investigating facts. Consider a Hotelling duopoly. Outlet 1 is a pay media outlet. We investigate the comparative statics of the profit maximizing choices of outlet 1 with respect to u2 ; for this exercise it does not matter whether outlet 2 is another commercial (pay or freely available) media outlet or a PSM. Example A1. Suppose that V1=[0, ¯ v] with 0 <¯ v< 1 /β , c1(v1)=kv2 1/ 2where k> 0is a parameter, and R1(v1)= 1 −βv1 with 0 <β< 1. The total audience has a fixed size normalized to 1. The market share of outlet 1is given by the Hotelling specification s1(v1,p1,u2)=     0, if 1 2+v1−p1−u2 2τ≤0, 1 2+v1−p1−u2 2τ,if 0<1 2+v1−p1−u2 2τ<1, 1, otherwise, where τ>0is a parameter for the degree of product differentiation. The profit of outlet 1is π1(v1,p1,u2)=s1(v1,p1,u2)(R1(v1)+p1)−kv2 1 2. Assume that 4kτ>(1−β)2(A1) in order that π1is strictly concave in (v1,p1)in the relevant range. Note that in this example k has to be sufficiently high for the second order condition to hold, hence the case of withholding facts is not a limit case of this example. In order to have a unified treatment of the cases of investigating facts (where k> 0) and withholding Games 2025,16, 21 19 of 46 facts (where k= 0), we will also comment on the case where inequality A1 is reversed in Appendix C.2. Before we proceed, we point out that Example A1 is strategically equivalent to a model where u1 is strictly concave in quality, and quality has linear costs. To see this, think of outlet 1 choosing w1:=kv1/ 2. Then u1=v1−p1=√2w1/k−p1 is strictly concave in w1 , and w1 has constant marginal costs of 1. A model with linear costs and concave consumer utility could result, for example, if quality represents the number of realizations from a noisy signal acquired by media firms, or the time spend on investigation before reporting. The concavity of consumer utility then represents decreasing marginal utility of new signal realizations or more time spend on investigating the same issue. Appendix C.1. Analysis of Example A1 Remark A1. In Example A1, suppose that the profit maximization problem of 1has an interior solution where 0 <v1<¯ v , p1> 0, R1> 0and 0 <s1< 1. 25 Then v1 and p1 are strictly decreasing in u2 .Moreover, u1=v1−p1 is strictly increasing in u2 if 2 kτ>(1−β)2 , and strictly decreasing if 2kτ<(1−β)2. Proof. In the relevant range, π1(v1,p1,u2)=1 2+v1−p1−u2 2τ(1−βv1+p1)−kv2 1 2. The partial derivatives are ∂π1 ∂p1=−1 2τ(1−βv1+p1)+1 2+v1−p1−u2 2τ, ∂π1 ∂v1=1 2τ(1−βv1+p1)−β1 2+v1−p1−u2 2τ−kv1. Moreover, ∂2π1 ∂p2 1 =−1 τ<0, ∂2π1 ∂v2 1 =−β τ−k<0, ∂2π1 ∂p1∂v1=1+β 2τ. Hence the determinant of the Hessian is 1 τβ τ+k−1+β 2τ2 >0 iff 4 kτ>(1−β)2 . This shows π1 is strictly concave in the relevant range if inequality (A1) holds. The first order conditions for an interior solution are 1 2τ(1−βv1+p1)=1 2+v1−p1−u2 2τ, 1 2τ(1−βv1+p1)=β1 2+v1−p1−u2 2τ+kv1. Games 2025,16, 21 20 of 46 Solving the first order conditions gives v∗ 1(u2)=(τ+1−u2)(1−β) 4kτ−(1−β)2, p∗ 1(u2)=(β(1−β)+2kτ)(τ−u2)+1−β−2kτ 4kτ−(1−β)2. Differentiate ∂v∗ 1(u2) ∂u2=−1−β 4kτ−(1−β)2<0, ∂p∗ 1(u2) ∂u2=−β(1−β)+2kτ 4kτ−(1−β)2<0. Moreover, from u∗ 1(u2)=v∗ 1(u2)−p∗ 1(u2), ∂u∗ 1(u2) ∂u2=2kτ−(1−β)2 4kτ−(1−β)2. Therefore, u∗ 1(u2) is strictly increasing in u2 if 2 kτ>(1−β)2 , and u∗ 1(u2) is strictly decreasing in u2if 2kτ<(1−β)2. It remains to check under which parameter constellations an interior solution exists. Note that v∗ 1(u2)>0 iff τ+1>u2, (A2) and p∗ 1(u2)>0 iff τ+1−β−2kτ (β(1−β)+2kτ)>u2. (A3) Note that 1−β−2kτ (β(1−β)+2kτ)<1 by inequality (A1). Thus inequality (A2) is implied by inequality (A3). We turn to advertising revenue next. Note that R1(v∗ 1(u2)) =1−β(τ+1−u2)(1−β) 4kτ−(1−β)2 is strictly positive iff u2>τ+1−4kτ−(1−β)2 β(1−β). (A4) Inequalities (A3) and (A4) hold simultaneously iff τ+1−β−2kτ (β(1−β)+2kτ)>u2>τ+1−4kτ−(1−β)2 β(1−β). (A5) By inequality (A1), the right hand side is strictly smaller than the left hand side; therefore (A5) is satisfied in a non-empty open set of values for u2. The requirement v1<¯ vis satisfied whenever ¯ vis sufficiently high. Finally, we need to make sure that 0 <s1u∗ 1(u2),u2<1. This is the case iff 0<1 2+v∗ 1(u2)−p∗ 1(u2)−u2 2τ<1, Games 2025,16, 21 21 of 46 or equivalently −τ<v∗ 1(u2)−p∗ 1(u2)−u2<τ. We have v∗ 1(u2)−p∗ 1(u2)−u2 =(τ+1−u2)(1−β) 4kτ−(1−β)2−(β(1−β)+2kτ)(τ−u2)+1−β−2kτ 4kτ−(1−β)2−u2 =τ2k−2kτ+(1−β)2−2ku2 4kτ−(1−β)2. Thus 0 <s1u∗ 1(u2),u2<1 iff −1<2k−2kτ+(1−β)2−2ku2 4kτ−(1−β)2<1, or equivalently −4kτ−(1−β)2<2k−2kτ+(1−β)2−2ku2<4kτ−(1−β)2. (A6) The expression in the middle is a strictly decreasing function of u2. Since u2<τ+1 by (A2), 2k−2kτ+(1−β)2−2ku2>2k−2kτ+(1−β)2−2k(τ+1) =−4kτ−(1−β)2, thus the first inequality in (A6) holds. Similarly, by (A4), 2k−2kτ+(1−β)2−2ku2 <2k−2kτ+(1−β)2−2k τ+1−4kτ−(1−β)2 β(1−β)! =1 β(1−β)β2−β+2k−β2+2β+4kτ−1. Therefore, a sufficient condition for the second inequality in (A6) is that 4kτ−(1−β)2−1 β(1−β)β2−β+2k−β2+2β+4kτ−1 =2(β(1−β)−k)4kτ−(1−β)2 β(1−β)>0, which is true iff β(1−β)>k. (A7) We have established that the problem has an interior solution under the conditions (A7), (A1), and (A5), which we repeat here for convenience: β(1−β)>k, 4kτ>(1−β)2, τ+1−β−2kτ (β(1−β)+2kτ)>u2>τ+1−4kτ−(1−β)2 β(1−β). Games 2025,16, 21 22 of 46 To see they can be satisfied simultaneously, first choose β and k such that the first line holds. Then choose τ such that the second line holds; note that depending on how you choose τ, either 2kτ>(1−β)2or 2kτ<(1−β)2. Finally, choose u2for the last line. A numerical example that satisfies all the constraints may be reassuring. Let β= 0.5, τ= 1.25, and u2= 1.5. For k= 0.11, 2 kτ= 2 ∗ 0.11 ∗ 1.25 = 0.275 >(1−β)2= 0.25 and u∗ 1(u2) is strictly increasing in u2 . For k= 0.09, 2 kτ= 2 ∗ 0.09 ∗ 1.25 = 0.225 < 0.25 < 4kτ=0.45, and u∗ 1(u2)is strictly decreasing. Within our parameter restrictions, u∗ 1(u2) is strictly increasing in u2 if k is large. An economic intuition is that the marginal costs of v1 are rapidly increasing if k is large, and hence then the falling price dominates the decrease in program quality. To give more details, recall that v∗ 1(u2) and p∗ 1(u2) are strictly decreasing in u2 . If k is large, the effect of u2 on v∗ 1(u2)becomes less important (smaller in absolute value): ∂ ∂k ∂v∗ 1(u2) ∂u2=4τ(1−β) 4kτ−(1−β)22>0. On the other hand, the effect of u2on p∗ 1(u2)also becomes less important: ∂ ∂k ∂p∗ 1(u2) ∂u2=∂ ∂k −β(1−β)+2kτ 4kτ−(1−β)2! =2τ1−β2 4kτ−(1−β)22>0 But note that 4τ(1−β)−2τ1−β2=2τ(1−β)2>0, thus ∂ ∂k ∂v∗ 1(u2) ∂u2 >∂ ∂k ∂p∗ 1(u2) ∂u2. That is, if k increases, the change of v∗ 1(u2) in u2 is vanishing quicker than the change of p∗ 1(u2) in u2 . For large enough k , u∗ 1(u2) increases in u2 because the falling price overcompensates for the falling quality. Appendix C.2. The Case Where Inequality (A1) Is Reversed This subsection considers the case where 4 kτ<(1−β)2 . This is of particular interest in order to have a unified treatment with the case of hiding information where k= 0. In this case, the profit maximization problem of 1 cannot have an interior solution where 0 <v1<¯ v , p1> 0, R1> 0 and 0 <s1< 1, since the necessary second order condition for a maximum would be violated. We continue to study pay media, i.e., we will focus on constellations where p1> 0 in the solution to the profit maximization problem; as we will show, a sufficient condition for this to be the case is that the competitor’s quality is not too high. Remark A2. Consider Example A1 but suppose that 4 kτ<(1−β)2 ,and the solution to the profit maximization problem of firm 1involves p1> 0, R1> 0and 0 <s1< 1. 26 If u2<τ− 1, then v1=¯ v and p1is strictly decreasing in u2. Proof. In the relevant range, π1(v1,p1)=1 2+v1−p1−u2 2τ(1−βv1+p1)−kv2 1 2. Games 2025,16, 21 23 of 46 Since in the profit maximum p1>0, the following first order condition has to hold: ∂π1(v1,p1) ∂p1=−1 2τ(1−βv1+p1)+1 2+v1−p1−u2 2τ=0. Solving for p1gives p1=p∗ 1(v1):=1 2(τ−u2+(1+β)v1−1). Note that the assumption that u2<τ−1 implies that p∗ 1(v1)>0 for all v1∈[0, ¯ v]. Define π1(v1):=π1(v1,p∗ 1(v1)) = 1 2+v1−1 2(τ−u2+(1+β)v1−1)−u2 2τ!1−βv1+1 2(τ−u2+(1+β)v1−1) −kv2 1 2. Differentiating π1(v1)shows that π′ 1(v1)=1 4τ(1−β)(τ−u2+v1−βv1+1)−kv1. Evaluated at v1=0, this is π′ 1(0)=1 4τ(1−β)(τ−u2+1)>0. Moreover, π′′ 1(v1)=1 4τ(1−β)2−k>0. It follows that in the profit maximum, v1=¯ v, and p1=1 2(τ−u2+(1+β)¯ v−1), which is strictly decreasing in u2. It remains to check there are parameter constellations where R1> 0 and 0 <s1< 1 in this solution. Note R1= 1 −β¯ v> 0 since ¯ v< 1 /β . Moreover, the market share of outlet 1 is 1 2+¯ v−1 2(τ−u2+(1+β)¯ v−1)−u2 2τ! =1 4τ(¯ v(1−β)+τ−u2+1)>0 since by assumption u2<τ− 1. Finally, the market share of outlet 1 is smaller than 100% iff ¯ v(1−β)−u2+1<3τ, which is true if τis sufficiently high. To relate Remark A2 to Lemma 2, note that here R′ 1(v1)=−β>− 1 for all v1 , therefore choosing the highest possible quality is optimal. Games 2025,16, 21 24 of 46 Appendix C.3. Summary To summarize, we found that in all cases considered in this Appendix, the price p1 is strictly decreasing in u2 . Moreover, the quality v1 is strictly decreasing in u2 unless k is small in which case v1 is constant in u2 . Finally, the utility of consuming outlet 1, u1=v1−p1 , is strictly increasing in u2 if k is small (see Remark A2) or k is large, but u1 is strictly decreasing in u2 if k is in an intermediate range (see Example A1). These predictions could, in principle, be tested empirically. Appendix D. Multidimensional Strategy Spaces This appendix provides a formal analysis of multidimensional strategy spaces. As in the main text, suppose that outlet i reports about ki topics, and let vi,k denote program quality of topic k . We assume that vi,k∈h0, vhigh i,ki with vhigh i,k> 0. Outlet i chooses a vector vi∈Vi=×ki k=1h0, vhigh i,ki . Consumer utility from consuming outlet i is ui=fi(vi) , where fi is continuous, strictly increasing with respect to each argument vi,k , and satisfies fi(0)= 0. Advertising revenue per consumer is Ri(vi) , where Ri:Vi→R+ is positive, continuous, weakly decreasing in each argument vi,k , and independent of v−i . The profit of i∈C is πi=si(ui,u−i)Ri(vi)−ci(vi) . Turning to the PSM, suppose that i∈P chooses vi∈Vi(bi) to maximize ui=fi(vi) subject to ci(vi)≤bi . As above, a higher budget may enlarge the feasible set Vi. Appendix D.1. Withholding Facts In the case of withholding facts, there are no direct costs of raising program quality. Thus, ci(vi)=0 for all i∈C. We show that our main results generalize. In our analysis, we make use of the fact that if the qualities vi of a commercial outlet i maximize its profit and generate utility ui for a consumer, then vi must maximize advertising revenue per consumer Ri subject to the constraint that the utility for the consumer is at least ui . (If not, then there exists a feasible ˆ vi that generates weakly higher utility, hence weakly higher demand, and at the same time generates strictly higher advertising revenue Ri , contradicting the optimality of the vi .) We can therefore decompose the problem of commercial media outlet i into two steps: The first step asks which vi maximizes advertising revenue subject to the constraint that the utility of the consumer is at least equal to some given ui. The second stage then optimizes over ui.27 Step 1 In the first step, the vector of program qualities is chosen to maximize advertising revenue per consumer, subject to the constraint that the utility of the consumer is at least equal to some given ui . The maximal value of advertising revenue under this constraint is R∗ i(ui)=max vi∈Vi{Ri(vi)|fi(vi)≥ui}. We show that R∗ i has all the features assumed about Ri in our main model (see Assumption 3) : R∗ i is positive, continuous, weakly decreasing in ui and independent of u−i. First, R∗ iis positive since Riis positive by assumption. Second, we use the Maximum Theorem to show that R∗ i is continuous. Ri is continuous by assumption. Let Ui:={ui∈R+|∃vi∈Vi:fi(vi)≥ui}. Since 0 ∈Vi and fi(0)= 0, 0 ∈Ui . Moreover, since Vi is compact and fi is continuous, by the Weierstrass Theorem a maximum achievable utility exists, thus Ui=0,maxvi∈Vifi(vi). Games 2025,16, 21 31 of 46 Appendix E.2. Large Spillover Effects: An Example In this appendix, we show by example that πi may have constant differences in (vi,v−i) if spillover effects are large. We consider a Hotelling duopoly. Outlet 1 is a commercial outlet. We investigate the comparative statics of the profit maximizing choices of outlet 1 with respect to v2 ; for this exercise it does not matter whether outlet 2 is another freely available commercial media outlet or a PSM. Example A3. Suppose that V1=R+, R1(v1,v2)=max{1−(αv1+βv2), 0} where α> 0and β> 0are exogenous parameters, s1 is given by a Hotelling specification. Moreover, suppose that the profit maximization problem of 1has an interior solution where v1> 0, R1> 0, and 0<s1<1.31 Note that R1in Example A3 satisfies Assumption A1. Remark A4. Consider Example A3. If α>β ,then π1 has strictly increasing differences in (v1,v2) in the relevant range. If α=β,then π1has constant differences in (v1,v2)in the relevant range. Proof. In the relevant range, s1(v1,v2)=1 2+v1−v2 2τ, hence ∂2s1(v1,v2) ∂v2∂v1=0 and ∂s1(v1,v2) ∂v1=−∂s1(v1,v2) ∂v2=1 2τ. The profit of commercial outlet 1 is π1(v1,v2)=s1(v1,v2)(1−(αv1+βv2)) −c1(v1). Hence ∂π1 ∂v1=1 2τ(1−(αv1+βv2)) −αs1(v1,v2)−∂c1(v1) ∂v1 and ∂2π1 ∂v1∂v2=α−β 2τ. Therefore, if α>β , π1 has strictly increasing differences in (v1,v2) . On the other hand, if α=β, then π1has constant differences in (v1,v2). An implication of Remark A4 is that, if the cost function c1 is strictly convex and twice differentiable, the profit maximizing program quality v∗ 1(v2) is strictly increasing in v2 if α>β, and v∗ 1(v2)is constant in v2if α=β.32 To conclude this appendix, we assume a quadratic cost function to illustrate that all the assumptions in Example A3 are consistent with each other. Suppose that c1(v1)=kv2 1/ 2, k>0. Then the best reply function is v∗ 1(v2)= 1 2τ−α 2+α−β 2τv2 α τ+k. Games 2025,16, 21 32 of 46 Example A3 assumed an interior solution with v1> 0, R1> 0 and s1∈(0,1) . To see these assumptions are consistent with each other, consider the symmetric case where both firms are commercial media and have the same cost and advertising revenue functions. In a symmetric equilibrium, v1=v2=1−ατ α+β+2kτ>0 iff ατ <1. Moreover, for i=1, 2, Ri(vi)=1−(α+β)(1−ατ) α+β+2kτ=τα2+αβ +2k α+β+2kτ>0. Finally, by symmetry s1(v1,v2)=s2(v1,v2)=1/2. Appendix F. Audience Functions with Decreasing Differences, the Logit and Nested Logit Models Appendix F.1. Audience Functions with Decreasing Differences: A Sufficient Condition for Proposition 1 Violations of Assumption 2 do not overturn our results when the elasticity of advertising revenue with respect to program quality is sufficiently high. To see this, note that the crucial inequality (1) in the proof of Proposition 1 holds if R′ i(vi) Ri(vi)≥ ∂2si(vi,v−i) ∂vj∂vi ∂si(vi,v−i) ∂vj . Under Assumption 2, the right hand side is negative hence the above inequality is always satisfied; when Assumption 2 is violated advertising revenue must react sufficiently strong to program quality for the inequality to hold. Appendix F.2. The Logit Model To illustrate, we consider a generalization of the logit model. Suppose there is mass of consumers normalized to one, and the market share of outlet iis si(vi,v−i)=fi(vi) ∑n+m j=0fjvj, where the functions fi(vi) are strictly positive and strictly increasing, and v0 is the utility of the outside option. We allow (but do not require) the functions fi to differ across media outlets. The logit model is a special case where fi(vi)=exp(µvi) for some exogenous parameter µ>0. Note that this audience function satisfies Assumption 1, but in general violates Assumption 2. In particular, if there are two or more commercial media outlets, si cannot have increasing differences for all i∈C , as we show below. We also prove, however, that a sufficient condition for Γbto be a supermodular game is that R′ i(vi) Ri(vi)≥f′ i(vi) fi(vi) for all i∈C . In the logit model, this sufficient condition reduces to R′ i(vi)/Ri(vi)≥µ for all i∈C. Games 2025,16, 21 33 of 46 This illustration shows that, while Assumption 2 is restrictive, decreasing differences in the demand functions do not necessarily overturn our results when advertising revenue reacts strongly on program quality. Appendix F.3. The Logit Model Violates Assumption 2 As in the last subsection, suppose that si(vi,v−i)=fi(vi) ∑n+m j=0fjvj, where the functions fi(vi) are strictly positive and strictly increasing, and v0 is the utility of the outside option. For k=i, ∂si ∂vk=−fi(vi)f′ k(vk) ∑n+m j=1fjvj2, ∂2si ∂vi∂vk=f′ k(vk)f′ i(vi)fi(vi)−∑j=ifjvj ∑n+m j=1fjvj3. This implies that, if there are two or more commercial outlets, si cannot have weakly increasing differences for all i∈C, so Assumption 2 is violated. Appendix F.4. A Sufficient Condition for Proposition 1 in the Logit Model Note that ∂2πi ∂vi∂vk=f′ k(vk)f′ i(vi)fi(vi)−∑j=ifjvj ∑n+m j=1fjvj3Ri(vi)−fi(vi)f′ k(vk) ∑n+m j=1fjvj2R′ i(vi) =f′ k(vk) ∑n+m j=1fjvj2  f′ i(vi)fi(vi)−∑j=ifjvj ∑n+m j=1fjvjRi(vi)−fi(vi)R′ i(vi)  >f′ k(vk) ∑n+m j=1fjvj2 f′ i(vi)−∑j=ifjvj ∑n+m j=1fjvjRi(vi)−fi(vi)R′ i(vi)  >f′ k(vk) ∑n+m j=1fjvj2−f′ i(vi)Ri(vi)−fi(vi)R′ i(vi) so a sufficient condition for πito have weakly increasing differences in (vi,v−i)is that −fi(vi)R′ i(vi)≥f′ i(vi)Ri(vi) or equivalently R′ i(vi) Ri(vi)≥f′ i(vi) fi(vi). In the logit model, fi(vi)=exp(µvi) and hence f′ i(vi)/f(vi)=µ . Therefore, our main results hold in the logit model whenever R′ i(vi)/Ri(vi)≥µfor all i∈C. Games 2025,16, 21 34 of 46 Appendix F.5. The Logit Model Satisfies Assumption (2 log) Here we establish that the logit model satisfies Assumption (2 log). Suppose there is mass of consumers normalized to one, and the market share of outlet iis si(vi,v−i)=fi(vi) ∑n+m j=0fjvj, where the functions fi(vi) are strictly positive and strictly increasing, and v0 is the utility of the outside option. We allow (but do not require) the functions fi to differ across media outlets. The logit model is a special case where fi(vi)=exp(µvi) for some exogenous parameter µ>0. Then ln si=ln(fi(vi)) −ln n+m ∑ j=0 fjvj! ∂ln si ∂vi=f′ i(vi) fi(vi)−f′ i(vi) ∑n+m j=0fjvj for j=i, this is strictly increasing in vj, thus ∂ ∂vj ∂ln si ∂vi >0. Therefore, ln sihas strictly increasing differences in (si,s−i). Appendix F.6. The Nested Logit Model Satisfies Assumption (2 log) This subsection considers the nested logit model, which is often used in empirical studies of media demand (see (Berry & Waldfogel,2015) for a survey). Our exposition of the nested logit model follows (Berry,1994). Firms are partitioned into groups. Suppose firm i belongs to group g . The market share of a firm i is given by si=si|gsg , where si|g is the share of firm iwithin its group g, and sgis the market share of group g. Moreover, si|g=expui 1−σ ∑k∈gexpuk 1−σ and sg=∑k∈gexpuk 1−σ1−σ ∑g′∑k∈g′expuk 1−σ1−σ, where σis a parameter with 0 ≤σ<1. We now show that the nested logit model satisfies Assumption (2 log). Remark A5. In the nested logit model, ∂2 ∂uj∂uiln si>0. Games 2025,16, 21 35 of 46 Proof. ln si=ui 1−σ−ln ∑ k∈g expuk 1−σ! +(1−σ)ln ∑ k∈g expuk 1−σ! −ln  ∑ g′ ∑ k∈g′ expuk 1−σ  1−σ   =ui 1−σ−σln ∑ k∈g expuk 1−σ! −ln  ∑ g′ ∑ k∈g′ expuk 1−σ  1−σ  . Differentiate with respect to ui, keeping in mind firm ibelongs to group g ∂ln si ∂ui=1 1−σ−σ 1−σ expui 1−σ ∑k∈gexpuk 1−σ −∑k∈gexpuk 1−σ−σexpui 1−σ ∑g′∑k∈g′expuk 1−σ1−σ. The sign of the crosspartial ∂2ln si ∂uj∂ui can be determined by considering how the terms in this sum depend on uj. Suppose j belongs to a different group than i . Then the first two terms are independent of uj , and the numerator of the third term is independent of uj as well. The sign of crosspartial is equal to the sign of ∂ ∂uj  −1 ∑g′∑k∈g′expuk 1−σ1−σ  >0. Suppose i and j belong to the same group g . Consider the terms one by one. The first term 1 /(1−σ) does not depend on uj . The second term is nondecreasing in uj . Consider the third term. Since σ≥ 0, ∑k∈gexpuk 1−σ−σ is nonincreasing in uj . Moreover, ∑g′∑k∈g′expuj 1−σ1−σ is strictly increasing in uj . This implies the third term is strictly increasing in uj. We conclude that ∂2ln si ∂uj∂ui>0. Appendix G. Entry: A Hotelling Example In this Appendix, we illustrate the our considerations on entry in a Hotelling model. Example A4. Assume that there are at most two media outlets active in the market. Conditional on entry and assuming that 0<si<1and vi∈h0, 1 βi,the profit of a commercial outlet i is πi=1 2+vi−vj 2τ(1−βvi)−F. Games 2025,16, 21 36 of 46 The fixed costs F> 0can be saved by staying out of the market. Let 3 βτ > 1 >βτ and F<βτ/2. We compare a mixed commercial and public media market (dual media market) with a purely commercial media market. In the dual market, there is one PSM with an exogenously given quality vP , and one commercial outlet decides whether to enter. In a purely commercial market, there is no PSM, and up to two commercial outlets may enter. We model entry by a standard two stage game, where in the first stage entry decisions are taken, and in the second stage qualities are chosen. The solution concept is subgame perfect equilibrium. The assumptions on parameters β , τ , and F allow us to focus on the most interesting cases. Specifically, the assumption 1 >βτ rules out situations where the profit maximizing quality equals zero. The assumption 3 βτ > 1 rules out situations where the best reply of a commercial outlet to a PSM with low quality is such that the commercial outlet has 100% market share. The assumption that F<βτ/ 2 ensures that in a purely commercial media market two media outlets enter. In the dual market, the commercial outlet will enter if the PSM’s quality is not too high. Define ˆ v:=1+βτ −p8βτF β. We will show below that ˆ v is the relevant cutoff for the PSM’s quality, below which the commercial outlet enters. Remark A6. Consider Example A4. In a purely commercial market, both commercial media outlets enter. (i) In a dual market where the PSM’s quality is vP<(1−βτ)/β ,the commercial outlet enters. The quality of the PSM and the quality of the commercial outlet are both strictly lower than the equilibrium qualities in a purely commercial duopoly. (ii) In a dual market where the PSM’s quality satisfies (1−βτ)/β<vP<ˆ v ,the private outlet enters. The qualities of the PSM and of the commercial outlet are both strictly higher than the equilibrium qualities in a purely commercial duopoly. (iii) In a dual market where the PSM’s quality is vP>ˆ v ,the private outlet will not enter. Proof. We solve the game by backward induction. Suppose there are two firms in the market. Firm i is a commercial outlet. Firm j may be a commercial outlet or a PSM. The first order condition of outlet iis ∂ ∂vi1 2+vi−vj 2τ(1−βvi)=1 2τ1−βτ −2βvi+βvj=0. The reaction function of iis v∗ ivj=1−βτ +βvj 2β>0 where the inequality follows from the assumption 1 >βτ.33 Note that v∗ ivjis strictly increasing in vj. Consider the purely commercial market. There are two symmetric potential entrants. If both of them enter, in equilibrium of the resulting subgame v∗ ivj=vj for i , j= 1,2, thus v1=v2=1−βτ β, and profits are πi=1 21−β1−βτ β−F=βτ 2−F. Games 2025,16, 21 37 of 46 Since by assumption F<βτ/2, in equilibrium both commercial firms enter. Now consider the dual market. Note that Ri(v∗ i(vP)) =1−β1+βvP−βτ 2β =1+βτ −βvP 2 is strictly positive iff vP<(1+βτ)/β . In case that vP≥(1+βτ)/β , i cannot generate a strictly positive revenue, and hence will not enter.34 For the rest of the proof, consider the case where vP<(1+βτ)/β , unless otherwise noted. The market share of firm iis si(v∗ i(vP),vP)=1 2+ 1+βvP−βτ 2β−vP 2τ =1+βτ −βvP 4βτ . Note this is positive when vP<(1+βτ)/β ; moreover siv∗ i(vP),vP is smaller than one (for all vP≥ 0 ) if 1 +βτ < 4 βτ or 1 < 3 βτ which we assume to be the case. The profit of firm iis πi(v∗ i(vP),vP)=(1+βτ −βvP)2 8βτ −F. The commercial outlet will enter if (1+βτ −βvP)2 8βτ >F, or equivalently vP<ˆ v:=1+βτ −p8βτF β. Note that the assumption F<βτ/2 implies ˆ v=1+βτ −p8βτF β> 1+βτ −q8βτ βτ 2 β=1−βτ β. We are now in a position to complete the proof. Case (i): vP<(1−βτ)/β . Then vP<ˆ v , thus the the commercial outlet enters. Moreover, vP is by assumption strictly lower than the equilibrium quality in a purely commercial duopoly (1−βτ)/β . Because of the strategic complementarities, vi is also strictly lower than the equilibrium qualities in a purely commercial duopoly. Case (ii): (1−βτ)/β<vP<ˆ v . Since vP<ˆ v , the private outlet enters. Moreover, vP is by assumption strictly higher than the equilibrium quality in a purely commercial duopoly (1−βτ)/β . Because of the strategic complementarities, vi is also strictly higher than the equilibrium qualities in a purely commercial duopoly. Case (iii): If ˆ v<vP<(1+βτ)/β , the commercial outlet will not enter because the revenue it can generate does not cover the fixed costs F . Moreover, as argued above, if vP≥(1+βτ)/β the commercial outlet cannot generate any strictly positive revenue and hence will not enter. Appendix H. Income Effects Consumers in our model have to pay taxes or licence fees to cover the budgets of the PSM. These payments are independent of individual media consumption. They could, Games 2025,16, 21 38 of 46 however, affect demand via income effects. Such income effects can strengthen our main results, however, when the media are normal goods, i.e., demand increases in income. Suppose that for i∈C , si(vi,v−i,b) is weakly decreasing in b (the higher b , the lower consumers’ remaining income; if media are normal goods, demand is lower). Moreover, suppose that si has weakly increasing differences in (vi,b) , i.e., demand reacts more on quality differences when income is lower. The strategic complementarities between the commercial media are not affected by the income effects. For i∈C and j∈P , consider the cross-partial ∂2˜ πi ∂bj∂vi= ∂2si(vi,v−i,b) ∂vj∂viRi(vi)+∂si(vi,v−i,b) ∂vjR′ i(vi)!¯ v′jbj +∂2si(vi,v−i,b) ∂bj∂viRi(vi)+∂si(vi,v−i,b) ∂bjR′ i(vi). The first line describes the effects studied in our main model above: an increase of bj increases ¯ vj and this has the effects studied above (the terms in the bracket are the same as in inequality (1) in the proof of Proposition 1). The second line stems from the income effect. Note that ∂2si(vi,v−i,b) ∂bj∂vi≥ 0 because si has weakly increasing differences in (vi,b) , and ∂si(vi,v−i,b) ∂bj≤ 0 because good i is normal; hence the second line is positive. This shows that income effects strengthen the strategic complementarities that drive our results. On the other hand, PSM might lead commercial media to exit the market. Income effects can strengthen this type of crowding out: the PSM do not only offer competing products, but also lower demand for commercial media via income effects. Appendix I. Equilibrium Uniqueness In this Appendix, we use the contraction approach to give sufficient conditions for equilibrium uniqueness in the main model with a linear audience function si , and in the case of withholding information with a logit audience function. Consider the case of a linear audience function first: Remark A7. Consider the case of a linear audience function si(vi,v−i)=ai+bivi−∑ j=i j∈C∪P cijvj. Assume that ai≥∑ j=i cij max vj∈Vj vj, bi>0, cij >0for all j =i. Moreover, assume that Ri is smooth with R′ i(vi)< 0for all vi∈Vi .A sufficient condition for a unique equilibrium is that 2bi≥∑ j=i cij, R′′ i(vi)≤0, and c′′ i(vi)≥0for all vi∈Vi with at least one of these inequalities strict. Games 2025,16, 21 39 of 46 The assumption ai≥∑j=icij maxvj∈Vjvj ensures that si≥ 0 for all (vi,v−i)∈Vi×V−i ; bi> 0 ensures that si is strictly increasing in vi ; cij > 0 ensures si is strictly decreasing in vj for j=i. Proof. With the linear audience function, ∂2si ∂v2 i =∂2si ∂vj∂vi=0, so sihas constant differences. It follows that ∂2πi ∂vi∂vj>0. Note that ∂πi ∂vi=bRi+siR′ i−c′ i and ∂2πi ∂v2 i =2bR′ i+siR′′ i−c′′ i<0 where the strict inequality follows because b>0>R′ iby assumption. The sufficient condition for a contraction is ∂2πi ∂v2 i +∑ j=i ∂2πi ∂vi∂vj <0. Here, ∂2πi ∂v2 i +∑ j=i ∂2πi ∂vi∂vj = 2bi−∑ j=i cij!R′ i+siR′′ i−c′′ i Therefore, a sufficient for uniqueness is that 2bi≥∑ j=i cij, R′′ i(vi)≤0, and c′′ i(vi)≥0 for all vi∈Vi with at least one of these inequalities strict. In the Hotelling or Spokes model in the relevant range where all market shares are interior, bi=∑j=icij, so 2bi>∑j=icij holds automatically. Remark A8. Consider the logit model si(vi,v−i)=exp(µvi) ∑n+m k=0exp(µvk) in the case of withholding information where ci(vi)= 0for all i .Moreover, assume that Ri is smooth, Ri(vi)> 0for all vi∈Vi ,and Ri is strictly log concave in vi .Then the game Γb is a parameterized supermodular game. Moreover, the equilibrium is unique. Games 2025,16, 21 40 of 46 Proof. Corollary 1 implies that Γb is a parameterized supermodular game. Note that πi> 0 for all (vi,v−i). Therefore, we can think of the commercial media outlets as maximizing ln πi=ln si(vi,v−i)+ln(Ri(vi)). Consider ln si=µvi−ln ∑ j expµvj!. Differentiate to obtain ∂ln si ∂vi=µ−µexp(µvi) ∑kexp(µvk). For j=i, the term µexp(µvi) ∑kexp(µvk)is strictly decreasing in vj, thus ∂ ∂vj ∂ln πi ∂vi=∂ ∂vj ∂ln si ∂vi >0. Therefore, ln πihas strictly increasing differences in (vi,v−i). Moreover, ln πi is strictly concave in vi . Since ln(Ri(vi)) is strictly concave in vi by assumption, this result follows from ∂2ln si ∂v2 i =−∑jexpµvjexp(µvi)µ2−exp(µvi)exp(µvi)µ2 ∑jexpµvj2 =−∑j=iexpµvjexp(µvi)µ2 ∑jexpµvj2<0. To prove uniqueness, we show that ln πisatisfies the contraction condition ∂2ln πi ∂v2 i +∑ j=i j=0  ∂2ln πi ∂vj∂vi <0. Here this is equivalent to ∂2ln si ∂v2 i +∂2ln Ri ∂v2 i +∑ j=i ∂2ln si ∂vj∂vi <0. By strict log concavity of Ri, it is enough to show that ∂2ln si ∂v2 i +∑ j=i ∂2ln si ∂vj∂vi≤0. From si=exp(µvi) ∑kexp(µvk) it follows that ln si=µvi−ln ∑ j expµvj!