Working Paper No . 239 Fictitious Play in Networks

This paper studies ctitious play in networks of noncooperative two-player games. We 5 show that continuous-time ctitious play converges to Nash equilibrium provided that the overall 6 game is zero-sum. Moreover, the rate of convergence is 1= , regardless of the size of the network. 7 In contrast, arbitrary n-player zero-sum games do not possess the ctitious-play property. As an 8 extension, we consider networks in which each bilateral game is strategically zero-sum, a weighted 9 potential game, or a two-by-two game. In those cases, convergence requires either a condition on 10 bilateral payo¤s or that the underlying network structure is acyclic. The results are shown to 11 hold also for the discrete-time variant of ctitious play, which entails a generalization of Robinsons 12 theorem to arbitrary zero-sum networks. Applications include security games, conict networks, and 13 decentralized wireless channel selection. 14 Keywords. Fictitious play networks zero-sum games conicts potential games Miyasawas 15 theorem Robinsons theorem 16 JEL-Codes C72 Noncooperative Games; D83 Search, Learning, Information and Knowledge, 17 Communication, Belief, Unawareness; D85 Network Formation and Analysis: Theory 18 19 *) Department of Economics, University of Zurich, Schönberggasse 1, CH-8001 Zurich, Switzerland; 20 christian.ewerhart@econ.uzh. ch. 21 **) Department of Economics, University of Zurich, Schönberggasse 1, CH-8001 Zurich, Switzerland; 22 kremena.valkanova@econ.uzh. ch. 23 1. Introduction 1 Fictitious play (Brown, 1949, 1951; Robinson, 1951) refers to a class of simple and intuitive models 2 of learning in games. The common element of such models is that a player is imagined to respond 3 optimally to an evolving belief on the behavior of his opponents, where the players belief at any point 4 in time is formed on the basis of the empirical frequencies of strategy choices made by his opponents 5 up to that point in time. Understanding the conditions under which ctitious play converges to Nash 6 equilibrium is important because such results help to clarify the intuition that equilibrium play may 7 be reached even if players are not perfectly rational.1 While variants of ctitious play are known 8 to converge in large classes of two-player games, the case of n-player games has been explored to a 9 much lesser extent.2 10 This paper studies the dynamics of ctitious play in general classes of network games. We 11 rst consider what we call zero-sum networks (Bregman and Fokin, 1987, 1998; Daskalakis and 12 Papadimitrou, 2009; Cai and Daskalakis, 2011; Cai et al., 2016). These are multiplayer zero-sum 13 games that can be represented as a network of two-player games. This class of games is actually quite 14 large and includes practically relevant examples of resource allocation games such as generalized 15 Blotto and security games. We show that the Lyapunov methods of Hofbauer (1995) and Harris 16 (1998) can be extended to the class of zero-sum networks. Speci cally, the Lyapunov function 17 considered in the present paper aggregates, across all players in the network, the maximum payo¤ 18 that could be obtained by optimizing against the empirical frequency distribution of prior play. 19 This function converges to zero at rate 1=t on any continuous-time ctitious-play (CTFP) path, and 20 regardless of the size of the network. In particular, CTFP converges to equilibrium in any zero-sum 21 network. However, as we also show with an example, arbitrary n-player zero-sum games do not 22 possess the ctitious-play property, i.e., the network assumption is crucial. 23 To gauge the role of the zero-sum assumption, we consider three additional classes of network 24 games. First, we look at networks of strategically zero-sum games, or conict networks.3 Thus, 25 The literature on ctitious play is too large to be surveyed here. For an introduction to the theory of learning in games, see Fudenberg and Levine (1998). The literature on learning in social networks has recently been surveyed by Acemoglu and Ozdaglar (2011). For a concise discussion of epistemic vs. dynamic foundations of Nash equilibrium, see Krishna and Sjöström (1997). Positive convergence results for n-players have been established, in particular, for games solvable by iterated dominance (Milgrom and Roberts, 1991), games with identical interests (Monderer and Shapley 1996b; Harris, 1998), and various classes of star-shaped network games (Sela, 1999). Jordan (1993) has shown that ctitious play need not converge in a non-zero-sum three-player game. See also Gaunersdorfer and Hofbauer (1996). Recent papers that look at conict networks include Bozbay and Vesperoni (2014), Dziubiński et al. (2016a), 1 each bilateral game in the network is assumed to be best-response equivalent in mixed strategies 1 to a zero-sum game. Moulin and Vial (1978) noted that ctitious play converges in this class of 2 two-player games. In networks of conicts, CTFP converges as well, provided that valuations in the 3 bilateral games satisfy a condition that we call pairwise homogeneity of valuations. This assumption is 4 satis ed, for example, in transfer networks considered by Franke and Öztürk (2015). When valuations 5 are heterogeneous, however, convergence need not hold in general. Intuitively, the aggregation of 6 bilateral payo¤s does not commute with the strategic equivalence because the payo¤ transformations 7 that turn two di¤erent bilateral games into zero-sum games need not be identical. We illustrate this 8 fact with a surprisingly simple example in the spirit of Jordan (1993). But convergence can still 9 be obtained with heterogeneous valuations when the underlying network structure is acyclic, i.e., a 10 disjoint union of trees.4 11 Next, we assume that bilateral games are weighted potential games. Applications include channel 12 selection problems in wireless communication networks, and the spreading of ideas and technologies 13 over social networks, for instance. Extending the analysis of Cai and Daskalakis (2011), it is shown 14 that CTFP converges to equilibrium in any network of exact potential games. However, as we show 15 with still another example, ctitious play need not converge in general networks of weighted poten16 tial games. Instead, in a somewhat unexpected analogy, the convergence result holds for weighted 17 potential games under the condition that the underlying network structure is acyclic. Finally, by 18 combining our ndings for conict networks with pairwise homogeneous valuations and for net19 works of exact potential games, we obtain a generalization of Miyasawas (1961) theorem to network 20 games on arbitrary graphs. As additional extensions, the paper looks at discrete-time ctitious play 21 (DTFP), generalizing Robinsons (1951) famous result for two-person zero-sum games to arbitrary 22 n-player zero-sum networks, and at the possibility of correlated beliefs which is a relevant aspect in 23 multiplayer games. 24 As a contribution of potentially independent interest, we substantially simplify the Lyapunov 25 approach to CTFP in two-player zero-sum games introduced by Hofbauer (1995) and Harris (1998). 26 That approach has traditionally combined the envelope theorem with the theory of di¤erential in27 Franke and Öztürk (2015), Huremovic (2016), Jackson and Nei (2015), König et al. (2015), Kovenock et al. (2015), amongst others. For a survey, see Dziubiński et al. (2016b). E.g., any star-shaped network considered by Sela (1999) is acyclic, but the network shown in Figure 1 below is not acyclic.