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Viable Nash Equilibria: Formation and Defection 1 VIABLE NASH EQUILIBRIA: FORMATION AND DEFECTION EHUD KALAI In memory of John Nash Abstract. Viable Nash equilibria play critical roles in the operation of social systems, whereas unviable Nash equilibria are often contrived, and have the appearance of useless theory. I study simple dual indices to assess the viability of an equilibrium : a formation index, F (), and a defection index, D(). Surprisingly, these simple indices (1) predict the performance of Nash equilia in social systems and lab experiments, and (2) uncover new properties of Nash equilibria and stability issues that escape current game theory re…nements. JEL Classi…cation Codes: C0, C7, D5, D9. 1. Overview Current economic analysis often relies on the notion of Nash equilibrium. Yet there are mixed opinions about the viability of this notion. On one hand many equilibria, referred to as viable in this paper, play critical rolls in func- tioning social systems and perform well in lab and …eld experiments. Unviable Nash equilibria, on the other hand, fail to perform in such experiments and Date: March 9, 2018, this version August 13, 2019. Key words and phrases. Normal form games, Nash equilibrium, Stability, Fault tolerance, Behavioral Economics. The author thanks the following people for helpful conversations: Nemanja Antic, Sunil Chopra, Vince Crawford, K…r Eliaz, Drew Fudenberg, Ronen Gradwohl, Yingni Guo, Adam Kalai, Fern Kalai, Martin Lariviere, Eric Maskin, Rosemarie Nagel, Andy Postlewaite, Larry Samuelson, David Schmeidler, James Schummer, Eran Shmaya, Joel Sobel and Peyton Young; and seminar participants in the universities of the Basque Country, Oxford, Tel Aviv, Yale, Stanford, Berkeley, Stony Brook Bar Ilan and the Hebrew University This paper replaces earlier versions of similar titles. 1 2 EHUD KALAI are not observed in functioning social systems.1 As a result, the uses of un- viable Nash equilibria in many recent theory and applied theory papers seem contrived, and have the appearance of useless theory. This paper focuses on the subject of equilibrium viability. As a starting point for a such a discussion, we study simple dual economic indices that suc- ceed in assessing equilibrium viability: A formation index, F (), that measure the di¢ culty of forming the equilibrium ; and a defection idex, D(), that measures the ability of a formed equilibrium to sustain defection. As dis- cussed in this paper, viable Nash equilibria are shown to perform well on these indices, whereas unviable equilibria perform poorly. The indices also identify new issues concerning Nash equilibrium stability that escape existing game theory re…nement.2 Two points about the approach are important: Viability assessment are subjective, and are better based on multi-valued indices instead of on binary game-theory re…nements. This is similar to the use of E(X) and SD(X) by an economic agent who tries to assess the viabil- ity of an investment X. While no index or re…nement tells her whether the investment X is viable or not, E(X) and SD(X) are useful tools for forming a subjective assessment of X’sviability. As we discuss in the paper, the sub- jective part of the assessments enables one to consider additional information such as history, context and objective of an equilibrium. Purposely, this paper takes a minimal departure from Nash equilibrium. It only replaces Nash assumption that "no opponent is a potential defector" by a single variable that represents the "number of opponents who are potential defectors." In doing so the paper identi…es the consequences that are due solely to the inclusion of this paricular variable. Moreover, this simple modi…cation means that the indices stay away from various re…nements of Nash equilibrium 1The literature on these issues is too large to survey here. Some key examples and references may be found in Smith (1982), Erev and Roth (1998), Crawford (1998), Kahneman and Tversky (2000), Goeree and Holt (2001), Camerer (2003) and many of their followup papers. 2A general survey of stability in game theory would be too large to discuss here. Some key examples include Aumann (1959), Selten (1975), Myerson (1978), Basu and Weibull (1991), Kreps and Wilson (1982), Kalai and Samet (1984), Kohlberg and Mertens (1986), Bernheim, Peleg, and Whinston (1987), Young (1993), Kandori, Mailath, and Rob (1993), Moreno and Wooders (1996), and Myerson and Weibull (2015). VIABLEEQUILIBRIA 3 and thus (1) remain relevant to a broad range of applications, and (2) rely on simple best-response computations that are comprehensible to most players. At the concluding section of the paper we point to applications that call for more re…ned viability indices. However, since such di¤erent re…nements may require di¤erent and inconsistent modi…cations to the notion of Nash equilibrium, their study is left for future research. 1.1. RELATED EARLIER LITERATURE. Low viability may be viewed as a non-binary re…nement of Nash equilibrium. As such, it presents concerns about stability that escape existing game theory re…nements. More specif- ically, the paper presents examples of equilibria that pass most re…nement tests, but would be considered of low viability in the context in which they are played. Due to their natural interpretations in strategic games, it is not suprizing that the indices appear in earlier game theoretic discussions. Formally, the D() index may be viewed as the level of subgame perfection3 in a two stage play of , in which the players are allowed to revise their chosen strategies after learning the choices of their opponents in stage one. Choosing in the …rst stage and playing the second stage with no revisions is k subgame perfect i¤ k < D().4 In applications, the D index has its roots in the computer science literature on distributive computing, see for example Ben-Or et al. (1988). Motivated by this literature, Eliaz (2002) provides an implemenation procedure for economic environments with faulty players. To do so, Eliaz introduces the concept of k-fault-tolerant equilibrium, and as we show in later sections of this paper, an equilibrium is k-fault-tolerant in the sense of Eliaz i¤ k < D(). 3see Kalai and Neme (1992) 4 More explicitly, in stage one every player i declares the strategy (pure or mixed) &i that she intends to play. In stage two, with full knowledge of the entire pro…le of intentions &, she chooses her actual strategy i = i(&). The payo¤ of the two-stage game is ui(&; ) = ui(), if her &i = i; otherwise ui(&; ) = w, where w is the lowest possible payo¤ of any player e e in . Consider the strategy pro…le = (; ) in which & = , and i (&) = &i. It follows directly from the de…nitions that is k-perfect i¤ k < D(). 4 EHUD KALAI In addition to Eliaz, the index D is the same as the level of resilience used by Abraham et al. (2006) in their implementation of secret sharing and dis- tributive computing games, and Gradwohl and Reingold (2014) develope a notion of fault tolerance that holds in equilibria of large games. The F index is new to this paper, but it is related to the earlier literature through its duality relationship with D, F = n D. However, the applications discussed in this paper go further than the ones above. While defections in this paper may be attributed to faulty players, as in the papers above, the analysis presented here deals also with rational players who are presented with incentives to defect. As a result, the indices are useful in addressing questions such as: when do rational players choose to defect, how to modify a game to turn potential defectors into loyalists and vice versa, how to form or switch an equilibrium in a game played by rational players, etc. 2. Viability Indices 2.1. GAME THEORETIC PERSPECTIVE. A game theory story that motivates the de…nitions of the indices below concerns n rational players about to play a focal strategy pro…le . However, each of these rational players is concerned that a number of her opponents may fail to play their strategies (despite their seeming optimality) and asks herself whether in view of her con- cerns her strategy is optimal. This is a di¢ cult question since there are many such concerns, and they are often di¢ cult to formulate. One concern is that some of her opponents are simply "faulty," irrational and unpredictable as described by Eliaz(2002). But there are also concerns she ascribes to ratio- nal opponents. Examples include pro…table defections by groups of rational opponents, outside bribes and threats applied to rational opponents, ratio- nal opponents desire to build reputation for future games, and many other concerns that were not included in the formal description of the game under consideration. Addressing the concerns above by standard Bayesian methods is impractical. It would require a listing of the large number of such concerns that each player VIABLEEQUILIBRIA 5 holds about each of her opponents, and assigning prior probability to each such concern. Notice, however, that these concerns become irrelevant when the player’s strategy is dominant in the game under consideration, since her strataegy is the best regardless of the strategies her opponents end up choosing. For this reason a pro…le that consists of all dominant strategies, i.e., a dominant strategy equilibrium, is most viable according to the indices of this paper, i.e., F () = 0 and D() = n. But when is not a dominant strategy equilibrium we may ask a related less demanding question: Against how many potential defectors are the players’strategies dominant? Such questions underly the de…nitions of D() and of F () below. 2.2. GAME TEORETIC NOTIONS. This paper focuses on a strategic n-person game = (N; A = i N Ai; u = (ui)i N ).
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