OPTIMISM AND PESSIMISM IN GAMES

OPTIMISM AND PESSIMISM IN GAMES Jürgen Eichberger Department of Economics, Universität Heidelberg, Germany David Kelsey Department of Economics University of Exeter, England University of Exeter October 2010 David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 1 / 29

But why did big shareholders not move to stop over-leveraging before it reached dangerous levels. Why did legislators not demand regulatory intervention? The answer I believe is that they had no sense of Knightian uncertainty. So they had no sense of the possibility of a huge break in housing prices and no sense of the fundamental inapplicability of the risk management models used in the banks. Risk came to mean volatility over the recent past.... Edmund Phelps, Financial Times, 12th May, 2009. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 2 / 29

INTRODUCTION Ambiguity refers to types of uncertainty for which it is di cult or impossible to determine the relevant probabilities. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 3 / 29

INTRODUCTION Ambiguity refers to types of uncertainty for which it is di cult or impossible to determine the relevant probabilities. Ambiguity has predictable comparative statics when there is strategic complementarity. If there are multiple equilibria an increase in ambiguity-aversion can cause the system to move to the lowest equilibrium. These issues appear relevant to the current nancial crisis. It was triggered by an increase in ambiguity concerning the returns of CDOs. The nancial system exhibits strategic complementarity. If one bank fails other banks are more likely to get into di culties. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 3 / 29

INTRODUCTION Ambiguity refers to types of uncertainty for which it is di cult or impossible to determine the relevant probabilities. Ambiguity has predictable comparative statics when there is strategic complementarity. If there are multiple equilibria an increase in ambiguity-aversion can cause the system to move to the lowest equilibrium. These issues appear relevant to the current nancial crisis. It was triggered by an increase in ambiguity concerning the returns of CDOs. The nancial system exhibits strategic complementarity. If one bank fails other banks are more likely to get into di culties. There appear to be multiple equilibria: David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 3 / 29

INTRODUCTION Ambiguity refers to types of uncertainty for which it is di cult or impossible to determine the relevant probabilities. Ambiguity has predictable comparative statics when there is strategic complementarity. If there are multiple equilibria an increase in ambiguity-aversion can cause the system to move to the lowest equilibrium. These issues appear relevant to the current nancial crisis. It was triggered by an increase in ambiguity concerning the returns of CDOs. The nancial system exhibits strategic complementarity. If one bank fails other banks are more likely to get into di culties. There appear to be multiple equilibria: one with a high level of optimism, activity and asset values; David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 3 / 29

INTRODUCTION Ambiguity refers to types of uncertainty for which it is di cult or impossible to determine the relevant probabilities. Ambiguity has predictable comparative statics when there is strategic complementarity. If there are multiple equilibria an increase in ambiguity-aversion can cause the system to move to the lowest equilibrium. These issues appear relevant to the current nancial crisis. It was triggered by an increase in ambiguity concerning the returns of CDOs. The nancial system exhibits strategic complementarity. If one bank fails other banks are more likely to get into di culties. There appear to be multiple equilibria: one with a high level of optimism, activity and asset values; one were traders are much more ambiguity-averse, economic activity and asset values are low. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 3 / 29

AMBIGUITY Uncertainty is ambiguous if the relevant probabilities are unknown or imperfectly known. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 4 / 29

AMBIGUITY Uncertainty is ambiguous if the relevant probabilities are unknown or imperfectly known. Situations which are clearly not ambiguous: Gambling David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 4 / 29

AMBIGUITY Uncertainty is ambiguous if the relevant probabilities are unknown or imperfectly known. Situations which are clearly not ambiguous: Gambling Most gambling devices are symmetric David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 4 / 29

AMBIGUITY Uncertainty is ambiguous if the relevant probabilities are unknown or imperfectly known. Situations which are clearly not ambiguous: Gambling Insurance Most gambling devices are symmetric David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 4 / 29

AMBIGUITY Uncertainty is ambiguous if the relevant probabilities are unknown or imperfectly known. Situations which are clearly not ambiguous: Gambling Insurance Most gambling devices are symmetric Insurance companies can get good estimates of probabilities from actuarial tables. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 4 / 29

AMBIGUITY Uncertainty is ambiguous if the relevant probabilities are unknown or imperfectly known. Situations which are clearly not ambiguous: Gambling Insurance Most gambling devices are symmetric Insurance companies can get good estimates of probabilities from actuarial tables. Economics Laboratories. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 4 / 29

We study games where there is ambiguity concerning the behaviour of one s opponents. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 5 / 29

We study games where there is ambiguity concerning the behaviour of one s opponents. We extend our previous work by allowing for ambiguity-preference instead of/as well as ambiguity-aversion. Evidence on individual choice under ambiguity. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 5 / 29

We study games where there is ambiguity concerning the behaviour of one s opponents. We extend our previous work by allowing for ambiguity-preference instead of/as well as ambiguity-aversion. Evidence on individual choice under ambiguity. Tort Law (Joshua Teitelbaum) David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 5 / 29

We study games where there is ambiguity concerning the behaviour of one s opponents. We extend our previous work by allowing for ambiguity-preference instead of/as well as ambiguity-aversion. Evidence on individual choice under ambiguity. Tort Law (Joshua Teitelbaum) Ambiguity preference causes people to exert too little care to prevent accidents. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 5 / 29

Goeree and Holt AER 2001 in Ten Little Treasures of Game Theory... David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 6 / 29

Goeree and Holt AER 2001 in Ten Little Treasures of Game Theory... Present ten games in which experimental evidence strongly supports Nash equilibrium. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 6 / 29

Goeree and Holt AER 2001 in Ten Little Treasures of Game Theory... Present ten games in which experimental evidence strongly supports Nash equilibrium. However, in each case, an apparently irrelevant parameter change produces results which contradict Nash equilibrium. In Are the Treasures of Game Theory Ambiguous?, (Economic Theory, forthcoming) we show that their evidence on normal form games can be explained by the hypothesis that subjects view their opponents behaviour as ambiguous. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 6 / 29

Goeree and Holt AER 2001 in Ten Little Treasures of Game Theory... Present ten games in which experimental evidence strongly supports Nash equilibrium. However, in each case, an apparently irrelevant parameter change produces results which contradict Nash equilibrium. In Are the Treasures of Game Theory Ambiguous?, (Economic Theory, forthcoming) we show that their evidence on normal form games can be explained by the hypothesis that subjects view their opponents behaviour as ambiguous. Ambiguity-preference formed an important part of this explanation. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 6 / 29

We present a new equilibrium concept, suitable for ambiguity-loving preferences. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 7 / 29

We present a new equilibrium concept, suitable for ambiguity-loving preferences. This uses a new notion of the support of a capacity. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 7 / 29

We present a new equilibrium concept, suitable for ambiguity-loving preferences. This uses a new notion of the support of a capacity. We prove existence of a pure equilibrium. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 7 / 29

We present a new equilibrium concept, suitable for ambiguity-loving preferences. This uses a new notion of the support of a capacity. We prove existence of a pure equilibrium. The paper studies the comparative statics of equilibrium. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 7 / 29

EXPERIMENTAL EVIDENCE Behaviour under ambiguity is di erent from behaviour under risk, e.g. in the Ellsberg Paradox; individuals under-weight likely events; unlikely events are over weighted; for a less familiar the source of uncertainty the under-weighting of likely events is greater and the over-weighting of unlikely events is less. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 8 / 29

DECISION WEIGHTS 1.... Weight 45 o 0 Likelilhood David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 9 / 29

CEU PREFERENCES We assume that players beliefs may be represented as capacities. De nition A function ν : P (S)! R is a capacity if, 1 ν(s) = 1, ν( ) = 0, 2 A B ) ν(b) ν(a). De nition Let ν be a capacity on S i. The dual capacity ν is de ned by ν (A) = 1 ν (:A) David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 10 / 29

An expected value with respect to a capacity may be de ned to be a Choquet Integral. Assume u (a (s 1 )) > u (a (s 2 )) >... > u (a (s n )). David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 11 / 29

An expected value with respect to a capacity may be de ned to be a Choquet Integral. Assume u (a (s 1 )) > u (a (s 2 )) >... > u (a (s n )). De nition The Choquet expected utility of u with respect to capacity ν is: Z u(a(s))dν(s) = u (a (s 1 )) ν (s 1 ) + where S = fs 1,..., s n g. n u (a (s k )) [ν(s 1,..., s k ) ν(s 1,..., s k 1 )], k=2 David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 11 / 29

An expected value with respect to a capacity may be de ned to be a Choquet Integral. Assume u (a (s 1 )) > u (a (s 2 )) >... > u (a (s n )). De nition The Choquet expected utility of u with respect to capacity ν is: Z u(a(s))dν(s) = u (a (s 1 )) ν (s 1 ) + where S = fs 1,..., s n g. n u (a (s k )) [ν(s 1,..., s k ) ν(s 1,..., s k 1 )], k=2 We shall assume that players preferences may be represented as maximising the Choquet expected value of utility with respect to a capacity. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 11 / 29

De nition If µ is a capacity on S : core(µ) = C(µ) = fp 2 (S) : p(a) µ(a), 8A Sg, where (S) is the set of additive probability distributions on S. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 12 / 29

De nition If µ is a capacity on S : core(µ) = C(µ) = fp 2 (S) : p(a) µ(a), 8A Sg, where (S) is the set of additive probability distributions on S. De nition A capacity µ is said to be convex if: µ(a [ B) > µ(a) + µ(b) µ(a \ B). David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 12 / 29

De nition If µ is a capacity on S : core(µ) = C(µ) = fp 2 (S) : p(a) µ(a), 8A Sg, where (S) is the set of additive probability distributions on S. De nition A capacity µ is said to be convex if: µ(a [ B) > µ(a) + µ(b) µ(a \ B). Proposition If µ is convex: Z u(a(s))d µ(s) = min p2c(µ) E pu(a), where C(µ) is the core of µ. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 12 / 29

We measure ambiguity by the maximal and minimal degrees of ambiguity of a capacity. De nition Let µ be a convex capacity on S i. De ne the maximal (resp. minimal) degrees of ambiguity of µ by: λ (µ) = max 1 µ (:A) µ (A) :? $ A $ S i, γ (µ) = min 1 µ (:A) µ (A) :? $ A $ S i. The maximal and minimal degrees of ambiguity provide upper and lower bounds on the amount of ambiguity which the decision-maker perceives. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 13 / 29

Ja ray-philippe Capacities De nition Say that ν is a Ja ray-philippe (JP) capacity if there exists a convex capacity µ and α 2 [0, 1], such that ν = αµ + (1 α) µ, where µ denotes the dual capacity of µ. If ν = αµ + (1 α) µ is a JP capacity then preferences can be represented in both the multiple priors and CEU forms Z fdν = α min E pf + (1 α) max E pf. p2c(µ) p2c(µ) David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 14 / 29

Ambiguity and Ambiguity-Attitude JP-capacities allow a distinction between ambiguity attitude as measured by α and ambiguity as measured by µ. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 15 / 29

Ambiguity and Ambiguity-Attitude JP-capacities allow a distinction between ambiguity attitude as measured by α and ambiguity as measured by µ. De nition Let ν = αµ + (1 α) µ and ν 0 = α 0 µ 0 + (1 α 0 ) µ 0 be two JP capacities. 1 We say that ν is more ambiguity-averse than ν 0 if α > α 0. 2 We say that ν is more ambiguous than ν 0 if for all A S, µ (A) 6 µ 0 (A). If ν is more ambiguous than ν 0 then C(µ 0 ) C(µ). David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 15 / 29

AMBIGUITY IN GAMES We consider a game Γ = hn; (S i ), (u i ) : 1 6 i 6 ni with nite pure strategy sets S i for each player and payo functions u i (s i, s i ). David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 16 / 29

AMBIGUITY IN GAMES We consider a game Γ = hn; (S i ), (u i ) : 1 6 i 6 ni with nite pure strategy sets S i for each player and payo functions u i (s i, s i ). De nition The payo function u i (s i, s i ) satis es increasing di erences in hs i, s i i if when ŝ i > s i, u i (ŝ i, s i ) u i ( s i, s i ) is increasing in s i. Increasing di erences says that if a given player plays a higher strategy, then his/her opponents have an incentive to increase their strategies as well. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 16 / 29

AMBIGUITY IN GAMES We consider a game Γ = hn; (S i ), (u i ) : 1 6 i 6 ni with nite pure strategy sets S i for each player and payo functions u i (s i, s i ). De nition The payo function u i (s i, s i ) satis es increasing di erences in hs i, s i i if when ŝ i > s i, u i (ŝ i, s i ) u i ( s i, s i ) is increasing in s i. Increasing di erences says that if a given player plays a higher strategy, then his/her opponents have an incentive to increase their strategies as well. De nition A game, Γ = hn; (S i ), (u i ) : 1 6 i 6 ni, has positive externalities if u i (s i, s i ) is increasing in s i for 1 6 i 6 n. Positive externalities and increasing di erences will be a maintained hypothesis. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 16 / 29

De nition of Equilibrium An equilibrium is a situation where players believe that their opponents will play best responses. However these are ambiguous beliefs. Hence a given player is not certain that his/her opponents will use equilibrium strategies. Players may react to ambiguity either optimistically or pessimistically. De nition A pair of capacities ν 1,ν 2 is an equilibrium under ambiguity if: 1 For all a 1 in the support of ν 1, a 1 maximises the expected utility of player 1 given that ν 2 represents player 1 s beliefs about the strategies of player 2. 2 For all a 2 in the support of ν 2, a 2 maximises the expected utility of player 2 given that ν 1 represents player 2 s beliefs about the strategies of player 1. If beliefs are additive this de nition coincides with Nash equilibrium. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 17 / 29

Support Notions Previous support notions are unsuitable for use when individuals may express ambiguity-preference. De nition The DW-support of the capacity ν, supp DW ν is a set E S i, such that ν (S i ne ) = 0 and ν (F ) > 0, for all F such that S i ne $ F. De nition The M-support of capacity ν, is de ned by supp M ν = fs 2 S i : ν (s) > 0g. Consider the neo-additive capacity ν (A) = δ (1 α) + (1 δ) π (A), where π is additive on S i. For δ > 0, α < 1, supp M ν = supp ν = S i. DW David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 18 / 29

Support of a JP-Capacity De nition Let ν = αµ + (1 by α) µ be a JP-capacity on S. We de ne the support of ν supp ν= \ supp π, π2c (µ) where C (µ) denotes the core of µ. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 19 / 29

De nition If ν is a capacity on S, de ne B (ν) = s 2 S : 8A $ S, s /2 A; ν (A [ s) > ν (A). B (ν) is the set of states given positive weight in the Choquet integral. In the Choquet integral of a given act a, the decision-weight assigned to state s is ν (fs : a (s) a ( s)g [ f sg) ν (s : a (s) a ( s)). David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 20 / 29

De nition If ν is a capacity on S, de ne B (ν) = s 2 S : 8A $ S, s /2 A; ν (A [ s) > ν (A). B (ν) is the set of states given positive weight in the Choquet integral. In the Choquet integral of a given act a, the decision-weight assigned to state s is ν (fs : a (s) a ( s)g [ f sg) ν (s : a (s) a ( s)). Proposition Let ν = αµ (A) + (1 α) µ (A), be a JP-capacity then supp ν B (ν). David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 20 / 29

Existence Of Equilibrium Theorem Let Γ be a game of positive externalities and increasing di erences. Then for any exogenously given n-tuples of ambiguity-attitudes α, maximal (resp. minimal) degrees of ambiguity λ (resp. γ), (γ 6 λ); the game Γ has a pure equilibrium ν = hν 1,...ν n i in JP-capacities, for 1 6 i 6 n. The maximal degree of ambiguity is at most λ i and minimal degree of ambiguity is at least γ i. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 21 / 29

Existence Of Equilibrium Theorem Let Γ be a game of positive externalities and increasing di erences. Then for any exogenously given n-tuples of ambiguity-attitudes α, maximal (resp. minimal) degrees of ambiguity λ (resp. γ), (γ 6 λ); the game Γ has a pure equilibrium ν = hν 1,...ν n i in JP-capacities, for 1 6 i 6 n. The maximal degree of ambiguity is at most λ i and minimal degree of ambiguity is at least γ i. Since the preferences we consider are not quasi-concave, traditional arguments based on xed point theorems will not work. Instead we use a lattice theoretic proof. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 21 / 29

COMPARATIVE STATICS I Multiple Equilibria Many games with strategic complements have multiple equilibria. Proposition Consider a game, Γ, of positive externalities with increasing di erences which satis es Assumption 5.1. There exist ᾱ (resp. α), 0 < α 6 ᾱ < 1, and γ such that if the minimal degree of ambiguity is γ (µ i ) > γ and α i > ᾱ, (resp. 6 α) for 1 6 i 6 n, equilibrium is unique and is smaller (resp. larger) than the smallest (resp. largest) equilibrium without ambiguity. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 22 / 29

COMPARATIVE STATICS II Ambiguity-Attitude For the second comparative static result we need the following assumption: De nition A game, Γ, has positive aggregate externalities if u i (s i, s i ) = u i (s i, f i (s i )), for 1 6 i 6 n, where u i is increasing in f i and f i : S i! R is increasing in all arguments. Players only care about a one-dimensional aggregate of their opponents strategies. Examples, Homogenous goods Cournot oligopoly. Private provision of a public good. Global pollutants e.g. CO 2. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 23 / 29

An increase in pessimism reduces the equilibrium strategies in games with positive aggregate externalities and strategic complements. We hold ambiguity constant by placing bounds on the maximal and minimal degrees of ambiguity. Theorem Let Γ be a game of positive aggregate externalities with increasing di erences. Let s (α) (resp. s (α)) denote the lowest (resp. highest) equilibrium strategy pro le when the minimal (resp. maximal) degree of ambiguity is γ (resp. λ). Then s (α) and s (α) are decreasing functions of α. Where α = hα 1,..., α n i denotes the vector of ambiguity attitudes. If there are multiple equilibria, the strategies played in the highest and lowest equilibria will decrease. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 24 / 29

EQUILIBRIUM q q q q - -.... David Kelsey (University of Exeter) OPTIMISM Figure: IN GAMES October 2010 25 / 29

The comparative statics are reversed in games of negative aggregate externalities; for further details see Eichberger & Kelsey (2002). David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 26 / 29

The comparative statics are reversed in games of negative aggregate externalities; for further details see Eichberger & Kelsey (2002). Ambiguity and ambiguity-attitude have distinct e ects. Increasing ambiguity causes the multiplicity of equilibria to disappear, while increasing ambiguity-aversion causes the equilibrium strategies to decrease. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 26 / 29

CONCLUSION We have extended our previous research (i.e. JET 2002) in the following ways. Larger Class of Preferences Ambiguity-loving as well as ambiguity-averse behaviour may be accommodated. Larger Class of Games Symmetry not assumed. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 27 / 29

CONCLUSION We have extended our previous research (i.e. JET 2002) in the following ways. Larger Class of Preferences Ambiguity-loving as well as ambiguity-averse behaviour may be accommodated. Larger Class of Games Symmetry not assumed. Do not assume pay-o functions are concave in own strategy. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 27 / 29

CONCLUSION We have extended our previous research (i.e. JET 2002) in the following ways. Larger Class of Preferences Ambiguity-loving as well as ambiguity-averse behaviour may be accommodated. Larger Class of Games Symmetry not assumed. Do not assume pay-o functions are concave in own strategy. Aggregate externalities not used except for comparative statics. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 27 / 29

EXPERIMENTAL EVIDENCE There is a limited amount evidence which tests these theories. The results are broadly positive. Colman & Pulford (2007) nd evidence of ambiguity in games, but do not test any particular theory. Mauro & Castro (2008) have a experiment on voluntary contributions to a public good. There are systematic deviations from Nash equilibrium; They explain these by a combination of ambiguity-aversion and ambiguity-preference. Eichberger et al. (2008) Subjects play either other subjects, a professor of game theory or the experimenter s grandmother. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 28 / 29

EXPERIMENTAL EVIDENCE There is a limited amount evidence which tests these theories. The results are broadly positive. Colman & Pulford (2007) nd evidence of ambiguity in games, but do not test any particular theory. Mauro & Castro (2008) have a experiment on voluntary contributions to a public good. There are systematic deviations from Nash equilibrium; They explain these by a combination of ambiguity-aversion and ambiguity-preference. Eichberger et al. (2008) Subjects play either other subjects, a professor of game theory or the experimenter s grandmother. The comparative statics of ambiguity are opposite in games of strategic complements and substitutes as predicted. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 28 / 29

EXTENSIONS Single Person Decisions David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 29 / 29

EXTENSIONS Single Person Decisions utility = u (a, s) if there are increasing di erences in ha, si, then an increase in ambiguity preference will cause the individual to choose a higher action. Games with Multiple Priors. Let C be a set of priors on S. We may de ne supp = \ supp π. π2c This can be used as the basis of a theory of games with multiple priors. The existence result and the rst comparative statics result could be extended to multiple priors in a straightforward way. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 29 / 29

EXTENSIONS Single Person Decisions utility = u (a, s) if there are increasing di erences in ha, si, then an increase in ambiguity preference will cause the individual to choose a higher action. Games with Multiple Priors. Let C be a set of priors on S. We may de ne supp = \ supp π. π2c This can be used as the basis of a theory of games with multiple priors. The existence result and the rst comparative statics result could be extended to multiple priors in a straightforward way. Further experimental tests. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 29 / 29

EXTENSIONS Single Person Decisions utility = u (a, s) if there are increasing di erences in ha, si, then an increase in ambiguity preference will cause the individual to choose a higher action. Games with Multiple Priors. Let C be a set of priors on S. We may de ne supp = \ supp π. π2c This can be used as the basis of a theory of games with multiple priors. The existence result and the rst comparative statics result could be extended to multiple priors in a straightforward way. Further experimental tests. Other Behavioural theories David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 29 / 29

EXTENSIONS Single Person Decisions utility = u (a, s) if there are increasing di erences in ha, si, then an increase in ambiguity preference will cause the individual to choose a higher action. Games with Multiple Priors. Let C be a set of priors on S. We may de ne supp = \ supp π. π2c This can be used as the basis of a theory of games with multiple priors. The existence result and the rst comparative statics result could be extended to multiple priors in a straightforward way. Further experimental tests. Other Behavioural theories Overcon dence. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 29 / 29

Assumption For 1 6 i 6 n, u i (s i, s i ) and u i (s i, s i ) have a unique maximizer, i.e. argmax si 2S i u i (s i, s i ) = 1 and argmax si 2S i u i (s i, s i ) = 1. Colman, A. & Pulford, B. (2007), Ambiguous games: Evidence for strategic ambiguity aversion, Quarterly Journal of Experimental Psychology 60, 1083 1100. Eichberger, J. & Kelsey, D. (2002), Strategic complements, substitutes and ambiguity: The implications for public goods, Journal of Economic Theory 106, 436 466. Eichberger, J., Kelsey, D. & Schipper, B. (2008), Granny versus game theorist: Ambiguity in experimental games, Theory and Decision 64, 333 362. Mauro, C. d. & Castro, M. F. (2008), Kindness confusion or... ambiguity, working paper, Universita of Catania. David Kelsey (University of Exeter) OPTIMISM IN GAMES October 2010 29 / 29