Utility, Risk And Nash Equilibrium Selection

Economic equilibria and equilibrium analytic are the main contents and analytic tools of economics. Since Adam Smith,most economic analyses have been done under perfect competition and they studied how economic agents influence the supply and demand of markets and how the equiibria be reached. The analyses for direct interactions between the agents are not enough. But direct interactions of agents are more important in many economic phenomena such as oligopoly,bid and etc. It is game theory that provides description methods and analytic tools for the need and obtains high recognition in economics circles. Three experts of game theory, J.Nash,R.Selten and J.Harsanyi,were awarded the Nobel economic prize of 1994 due to this reason. The broad applications of game theory in economic fields are remodeling the economics. In the meantime,game theory,formerly a branch of mathematics,is also becoming an economic branch. " Economic game theory * and " economics of game theory " appear in many economic literatures.Nash equilibrium is fundamental concept of game theory. Harsanyi and Selten (1988) pointed that it has three weaknesses: multiplicity of equilibria, unstablity of proper mixed Nash equilibrium and imperfect equilibria. Among the three, multiplicity of equilibria is crucial,it cause uncertainty,that is,which strategy players (or decider) should selected is still unknown. In order to solve this question,an important research field of game theory—the refinement and selection of Nash equilibria—has been established.Subgame perfect Nash equilibrium (Selten, 1965) jtrembling hand equilibrium (Selten,1975)and proper equilibrium (Myerson,1978) are new concepts for refinement,but the results are unsatisfactory ( unique equilibrium not reached ).Kohlberg and Merten (1986 ) proposed idea of stable equilibria and argued that there is no one equilibrium to satisfy all the requirements.Harsanyi and Selten(1988) also worked out equilibrium selection methods—payoff domina-tion and risk domination analyses,but they were often violated by experimental evidence(Cooper et al. 1990,Huyck et al. 1990,1991).According to Nash's definition,many games have more than one equilibria,but players select only one strategy (or equilibrium) in practical decisions. Our paper argues that the difficulties of equilibrium refinement and selection are due to Nash's definition. In the definition,Nash did not point out payoff is player's utility or income. In economic analyses it should be player's utility not income,but utility cannot become players' common knowledge (even in games of complete information). Therefore,our paper first models economic problem as income model of game (players' incomes could be common knowledge in games of complete information),then using utility functions establishes utility model of game.So the properties and selection of Nash equilibria are related with the utility characteristics of player (such as risk aversion,risk seeking and etc).Our paper are divided as four chapters.Chapter one analyses the risk revenue-the cost economic agents pay for transforming risk asset (random variable) into deterministic one (the mean ),through which we can compare deterministic quantities to determine the preferences among risk assests.Chater two first introduces utility functions to establish utility model of game,and then discusses the relation between properties of equilibrium and utility functions.Chapter three discusses relation between the strategy or equlibrium selection and player's risk characteristics. Chapter four tests the models proposed in chapter three through some experiments.In chapter one,we establish an dynamic utility model to describe uncertainty.TheoremO.l // (l)g{z) : B* -> JR satisfies:(a)3C > O^z1) - g(z2)\ < C\zl - z2\ holds,Vz\z2 e B,. (b) g{0) = 0, (2)1c € D,aggregator f(ct,v) : itf x [0, T] xftx R ->? B: satisfies Lip condition in v,and E ff \f(ct, v)\2ds < oo,(3) u(.) is increasing and bounded, thenVc € D,there is a unique square-integral process {Vf}o aVt(c) + (1 - a)Vt(c), Vt € [0,1] (9.3)holds,then economic agent is uncertainty aversion.Theorem 0.2 If conditions of theorem 0.1 hold, and for any (c, v) € 1$+1, Hessian matrixes D2u, ^3'"* of u(c), f{c,v) in c are negative semidefinite,g(z) — —n\z\, then economic agent is uncertainty aversion.Theorem 0.3 If conditions of theorem 0.1 hold andu(c),f{c,v) are increasing in c,for two consumption processes c,c € D,ifct > c't holds]i{t,w) € [0,T] x ft , then Vt(c) > V*(c'),Vt € [0,T];and if P{cT > c?) > 0 also holds,then strict inequality holds.Now we consider asset {X(}o<* 0,u"(.) < 0,then * € [0, T] (0.6)(2)ifu satisfies .V(.) > 0,u"(.) > 0,then > u(eg(X\?t)), t € [0,T] (0.7)For g-EU model of asset,we can define uncertainty premium of asset X. Definition 0.3 Tx = Fx is called uncertainty premium of asset X,wherertx=e,{X\Ft)-u-l{Vt{X)) (0.8)Using Ito formula, we can derive that u-1(Vj(X)) is the solution of BSDE:Yt = X + [Tg{s)ds - fTzsdWs, t 6 [0,T] Jt Jtwhere,1 u~l"(Vt)~9{t) = 9i'Zt) ~ 2^*zt = u-l'(Vt)ztu~l{Vt(X)) is recursive, so Px = eg(X\Jrt) - u~l{Vt(X)) is also recursive.Theorem 0.5 For two bounded utility functions ui(.) , u2(-) > tfr1^) > r2(.) 1 , then for any T-measurable asset X , rx* > T2^ holds,Vt € [0, T]Theorem 0.6 // utility function u is bounded and satisfies t u'(.) > 0 ) u"(.) < 0 , and absolute risk aversion r(.) is decreasing , then for any T-measurable asset X and constant c > 0 , Tx+C < Tx holds fort € [0, T] .The uncertainty in g-EU model depends on Lip constant fi,, because fi decides the sizes of sets E and ri.r(.) = ~wT is Arrow-Pratt measure of absolute risk aversionTheorem 0.7 In two g-EU models , if Lip constants fix > H2,then for any economic agent (his utility function is u(.)) and an asset X € L2{0,,!F,V, M),holds, where,T]f, Y*x are uncertainty premiums of asset X in two g-EU models.Risk premium of expectation utility model and uncertainty premium of RDEU are special cases of uncertainty premium of g-EU model in static setting.Their special properties are discussed in appendix 1 and 2.In chapter two,we describe economic problems as income model and utility model of game. Income model of game G = (N,(Si)ieN,(ai)ieN) is composed of set of players,strategy sets 5,-(t € N) and income functions a,(f € N). Nash equilibrium of income model can be defined on G.Definition 0.4 Strategy combination a* = (erf,... , Eaiio-i, a*_?), V (u?)?€Jv)>wnere u?(* ￡ N) is expectation or non-expectation utility functions,and suppose that it satisfied (1) ( Frechet ) differentiate, (2)first-order stochastic dominance preference in terms of income a,-( Ui(ai( (a22 - ai2)(622 - &21) (0.12)holds.Definition 0.7 Equilibrium (UUU2) payoff-dominates (Vl,V2))ifan > a22622.v\vPlayer2 Player l\au, bnan. bna2\, KFigure 0.1Harsanyi ,Selten (1988) pointed out that in the game with two strict pure strategy equilibria,player should first consider payoff-dominance and select payoff-dominant strategy ( or equilibrium),and when there is no payoff-dominance relation,he considers risk-dominance and selects risk-dominant strategy (or equilibrium). Introducing expectation utility functions,we will find that selection of strategy and equilibrium by player,who pursues maximum of expectation utility,is correlative with risk characteristics of him (risk aversion or risk seeking). Conse-quently,Harsanyi and Selten's theories are modified.Suppose ?i(.),u2(.) are expectation utility functions of playerl,2,and they are twince continuously differentible and strictly increasing. Introducing utility functions,the game depicted by figure 0.1 transforms into figure 0.2. According to Harsanyi and Selten's preliminary expectation theory (Harsanyi, Selten,1988 ),equilibrium (1/1,1/2) is selected if and only if:"l(on) -"1(021) "2(622)-"2(^1) ,n...—7—\------7—\ > —7T~i------7r~T (0.13)"lVa22J ~" "lla12J "2(,01lJ "~ "2(^12/Suppose that player infers characteristics of the other player according to those of himself,player l's anticipation of player 2's utility function be written as u\(.) and player 2's anticipation of player l's utility function as u|(.). From player l's opinion,he selects strategy U\ if and only if:Man) - ut(a2l)"1(022) - "1(012) ?!(6h) - ?f(6i2)(0.14)From player 2's opinion,he selects strategy U2 if and only if: 1*2(011) — ^2(021) ^2(^22) — t*2(&2i)(0.15)\Player2 Playerl\u2".KW^l.)Ml(flu)? "2(^2)"l(a2lX "2(*2l)"l (^22 ), ?2(MFigure 0.2Therefore,equilibrium (Ui^Uj) of game depicted by figure 0.2 is selected if and only if inequality (0.8),(0.9) hold.We analysis two kinds of 2 x 2 games. The first one is the game which satisfies an > <*2i > <*22 > ai2>&n > 612 > 622 > &21 and inequality (0.12),written as game (G-l),where (C/i, C/2) risk-dominates and payoff-dominates (Vi, V2). We obtain the following conclusions.Theorem 0.111. If player 1,2 are risk-seeking,equilibrium (C/x, C/2) of the game (G-l) is selected.2.If one or two of player 1,2 are risk averse,then equilibrium (U\, U2) or (Vi, V non- equilibrium (U\,V an > an > 021,^22 > &21 > &11 > 612 and inequality (0.12),written as(G-2),where (t/1? f/2) risk-dominates (Vi, V2),but is payoff-dominated by (Vi, V2).The conclusions about this kind of games as following:Theorem 0.12 ■ 1. If player 1,2 are risk-averse,equilibrium (i/i,^) w selected.2. If one or two of player 1,2 are risk-seeking,then equilibrium (1/1,1/2) or(Vi, V2),even non-equilibrium (U\, V2) or (Vi,U2) can be caught out. When both of them are risk-seeking and their Arrow-Pratt risk-seeking measures large enough,then equilibrium (Vi,V2) should be selected.Through the discussion of this chapter,we receive the conclusion: player's selection of strategy or equilibrium is correlated with the characteristics of himself and the types of game,and risk-dominance defined by Harsanyi and Selten is not really risk comparison of strategies or equilibria because risk-averse player selects risk-dominated strategy in game (G-l).Although the games discussed here are only 2x2 ones , according to transitive property of player's preference , we can generalize the conclusions to more general games which have more than two equilibria.Chapter four uses experimental evidence to test the theory proposed in chapter three .For the 2x2 coordination game depicted by figure 0.1 (atJ = bji,(i,j = 1,2)),player 1 select strategy U\ if and only if:ul(0l) uV(m) (a22 ~ (au-a21)(bn-bl2) {V- ]where 9X € [a2i,an],d2 € [a12,a22],?7i € [612,611] and r)2 € [621,622].Definition 0.8 R^,, is called risk-dominant ratio of equilibrium (UltU2) to(Vi,V2),ifp _ (Qn(a22 -It can be proved that R^y > Q,RUV > 1 <=>? (Ui,U2) risk-dominates (Vi, V2),and ^ < 1 ?=>? (Uu U2) is risk-dominated by (Vi, V2).Definition 0.9 Puv is called payoff-dominant measure of equilibrium (￡/i,(/2) to(VuV2),ifPuv = min(an -It can be easily seen that if Puv > 0,then (Ui, U2) payoff-dominates (Vi, V2).The left side of inequality (0.10) has relations with player's utility function,it depends on convexity or concavity of utility function,that is the player's attitude toward risk.Therefore,we propose three hypotheses:Hypothesis Lplayer selects payoff-dominant (equilibrium) strategy in 2x2coordination games.HypolhesisILthe selection of strategy or equilibrium in 2x2 coordination games is correlated with risk-dominant ratio of the strategy.HypothesisIILthe probability of strategy or equilibrium selected by oneplayer is correlated with his attitude toward risk.In order to test the hypotheses,we design 4 games (figure 0.3) and 5 questions about player's attitude toward risk. About 200 graduate and undergraduate students of economics college participated in the experiments and were asked to answer 5 questions. According to x2-test,Hypothesis I was refused and Hypothesis II was supported by the experimental evidence,and the probability of strategy selected was positively correlated with its risk-dominant ratio.Game 1Game 2"\Player 2 Player 1 N^v210,101,99,18,8Game 3""\Player 2 Player 1 >^Vr10,101, 6VK6,13, 3"\Player 2 Player 1^\^u2v210, 101, 7vx7, 14.5, 4.5Game 4"\Player 2 Player 1^\u2v210, 101, 6vx6, 14.5, 4.5Figure 0.3In order to test the Hypotheses HI , according to answers of 5 questions we divided the players into 3 parts of risk-seeking tendency,risk-averse tendency and no tendency,which have 56,65 and 71 persons respectively. Using contingency table,weobtained the x2-statistics of the four games :1.52(game l),,1.72(game 2),5.62(game 3) and 8.76(game 4). At the significant level 0.05,only the experimental results of game 3 support Hypotheses III,but at the significant level 0.10,the experimental results of game 3 and 4 support Hypotheses III. Although the experimental evidence from game 1 and 2 does not supported Hypotheses III,they do not violate the theory proposed in chapter 3. According to general economic hypothesis,players should be risk averse,the types of risk tendency only distinguish the extent of player's risk aversion. Game 1 and 2 belong to second kind of games of chapter 3,only when players are risk-seeking and the extent of risk-seeking are high enough,the strategies they selected change. Therefore evidence from these two games does not show the relation between strategy selection and player's risk tendencies. This just accords with the theory of last chapter.In general,the experimental evidence supports the theory of chapter 3.