Testing for Ambiguity in Coordination Games
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1 Testing for Ambiguity in Coordination Games David Kelsey Department of Economics, University of Exeter, England Sara LeRoux Department of Economics Oxford Brookes University University of Exeter. September 2014 David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
2 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.) Ambiguity in Coordination Games September / 17
3 INTRODUCTION Ambiguity refers to types of uncertainty for which it is di cult or impossible to determine the relevant probabilities. A large experimental literature has shown that ambiguity is an important in uence on decision-making. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
4 INTRODUCTION Ambiguity refers to types of uncertainty for which it is di cult or impossible to determine the relevant probabilities. A large experimental literature has shown that ambiguity is an important in uence on decision-making. However most of this literature concerns single person decisions. There are relatively few experiments which test for ambiguity in games. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
5 INTRODUCTION Ambiguity refers to types of uncertainty for which it is di cult or impossible to determine the relevant probabilities. A large experimental literature has shown that ambiguity is an important in uence on decision-making. However most of this literature concerns single person decisions. There are relatively few experiments which test for ambiguity in games. In the present paper we test how ambiguity a ects behaviour in games. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
6 INTRODUCTION Ambiguity refers to types of uncertainty for which it is di cult or impossible to determine the relevant probabilities. A large experimental literature has shown that ambiguity is an important in uence on decision-making. However most of this literature concerns single person decisions. There are relatively few experiments which test for ambiguity in games. In the present paper we test how ambiguity a ects behaviour in games. Games provide a link to economic models. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
7 INTRODUCTION Ambiguity refers to types of uncertainty for which it is di cult or impossible to determine the relevant probabilities. A large experimental literature has shown that ambiguity is an important in uence on decision-making. However most of this literature concerns single person decisions. There are relatively few experiments which test for ambiguity in games. In the present paper we test how ambiguity a ects behaviour in games. Games provide a link to economic models. In games subjects create ambiguity for one another rather than the experimenter creating ambiguity. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
8 Results Summary We nd strong evidence of ambiguity-aversion in games. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
9 Results Summary We nd strong evidence of ambiguity-aversion in games. Equilibrium Under Ambiguity (EUA see Eichberger, Kelsey, and Schipper (2009)) predicts play in these games better than Nash equilibrium or iterated dominance. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
10 Results Summary We nd strong evidence of ambiguity-aversion in games. Equilibrium Under Ambiguity (EUA see Eichberger, Kelsey, and Schipper (2009)) predicts play in these games better than Nash equilibrium or iterated dominance. Ambiguity has a larger impact in games than in urn experiments. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
11 Results Summary We nd strong evidence of ambiguity-aversion in games. Equilibrium Under Ambiguity (EUA see Eichberger, Kelsey, and Schipper (2009)) predicts play in these games better than Nash equilibrium or iterated dominance. Ambiguity has a larger impact in games than in urn experiments. Ambiguity-attitudes appear to be context dependent. Attitudes in games tend to di er from ambiguity-attitudes in urn experiments. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
12 Ambiguity Theory We use the neo-additive model of ambiguity, axiomatised by Chateauneuf, Eichberger, and Grant (2007). They represent preferences by: αδm (a) + δ (1 α) m (a) + (1 δ)e π u (a), (1) M (a) denotes the maximum utility of act a, m (a) denotes the minimum utility of act a, E π u (a) denotes the expected utility of act a. This is a special case of the Choquet expected utility model, Schmeidler (1989), which represents beliefs as capacities. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
13 Ambiguity Theory We use the neo-additive model of ambiguity, axiomatised by Chateauneuf, Eichberger, and Grant (2007). They represent preferences by: αδm (a) + δ (1 α) m (a) + (1 δ)e π u (a), (1) M (a) denotes the maximum utility of act a, m (a) denotes the minimum utility of act a, E π u (a) denotes the expected utility of act a. This is a special case of the Choquet expected utility model, Schmeidler (1989), which represents beliefs as capacities. δ is a measure of perceived ambiguity; David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
14 Ambiguity Theory We use the neo-additive model of ambiguity, axiomatised by Chateauneuf, Eichberger, and Grant (2007). They represent preferences by: αδm (a) + δ (1 α) m (a) + (1 δ)e π u (a), (1) M (a) denotes the maximum utility of act a, m (a) denotes the minimum utility of act a, E π u (a) denotes the expected utility of act a. This is a special case of the Choquet expected utility model, Schmeidler (1989), which represents beliefs as capacities. δ is a measure of perceived ambiguity; α measures ambiguity-attitude, α = 1 (resp. α = 0) corresponding to pure optimism (resp. pessimism). David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
15 Ambiguity Theory We use the neo-additive model of ambiguity, axiomatised by Chateauneuf, Eichberger, and Grant (2007). They represent preferences by: αδm (a) + δ (1 α) m (a) + (1 δ)e π u (a), (1) M (a) denotes the maximum utility of act a, m (a) denotes the minimum utility of act a, E π u (a) denotes the expected utility of act a. This is a special case of the Choquet expected utility model, Schmeidler (1989), which represents beliefs as capacities. δ is a measure of perceived ambiguity; α measures ambiguity-attitude, α = 1 (resp. α = 0) corresponding to pure optimism (resp. pessimism). Only 2 additional parameters needed compared to expected utility. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
16 The Games Player 1 Player 2 L M R T 0, 0 300, , x B 100, 300 0, 0 55, x x = 230, 120, 200, 170, 260, 60 (in that order). This is the familiar Battle of the Sexes game modi ed by adding an extra secure option for player 2. We believe subjects are likely to perceive ambiguity in the Battle of the Sexes since. It has two pure strategy Nash equilibria. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
17 The Games Player 1 Player 2 L M R T 0, 0 300, , x B 100, 300 0, 0 55, x x = 230, 120, 200, 170, 260, 60 (in that order). This is the familiar Battle of the Sexes game modi ed by adding an extra secure option for player 2. We believe subjects are likely to perceive ambiguity in the Battle of the Sexes since. It has two pure strategy Nash equilibria. Our experiments were one shot. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
18 Nash Equilibrium Theorem The game has the following Nash equilibria: 1 When 0 x 75, there are 3 equilibria: ht, Mi, hb, Li and 1 4 T B, 1 4 L M. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
19 Nash Equilibrium Theorem The game has the following Nash equilibria: 1 When 0 x 75, there are 3 equilibria: ht, Mi, hb, Li and 1 4 T B, 1 4 L M. Note R is dominated by the mixed strategy 1 4 L M. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
20 Nash Equilibrium Theorem The game has the following Nash equilibria: 1 When 0 x 75, there are 3 equilibria: ht, Mi, hb, Li and 1 4 T B, 1 4 L M. Note R is dominated by the mixed strategy 1 4 L M. 2 When D 75 < x 100, there are E3 equilibria: ht, Mi, hb, Li and x 100 T + (100 x ) 100 B, 1 61 M R. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
21 Nash Equilibrium Theorem The game has the following Nash equilibria: 1 When 0 x 75, there are 3 equilibria: ht, Mi, hb, Li and 1 4 T B, 1 4 L M. Note R is dominated by the mixed strategy 1 4 L M. 2 When D 75 < x 100, there are E3 equilibria: ht, Mi, hb, Li and x 100 T + (100 x ) 100 B, 1 61 M R. 3 When 100 < x < 300, there is a unique equilibrium: hb, Li, which also satis es iterated deletion of dominated strategies. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
22 Equilibrium Under Ambiguity (EUA) Theorem The game has the following Equilibria under Ambiguity: 1 When 0 x (1 δ)75, there are 3 equilibria: ht, Mi, hb, Li and 1 4 T B, 1 4 L M. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
23 Equilibrium Under Ambiguity (EUA) Theorem The game has the following Equilibria under Ambiguity: 1 When 0 x (1 δ)75, there are 3 equilibria: ht, Mi, hb, Li and 1 4 T B, 1 4 L M. 2 When (1 δ)75 < xd (1 δ)100, there are 3 equilibria: ht, Mi, hb, Li and x (1 δ)100 T + [(1 δ)100 x ] (1 δ)100 B, 1 61 M R E. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
24 Equilibrium Under Ambiguity (EUA) Theorem The game has the following Equilibria under Ambiguity: 1 When 0 x (1 δ)75, there are 3 equilibria: ht, Mi, hb, Li and 1 4 T B, 1 4 L M. 2 When (1 δ)75 < xd (1 δ)100, there are 3 equilibria: ht, Mi, hb, Li and x (1 δ)100 T + [(1 δ)100 x ] (1 δ)100 B, 1 61 M R E. 3 When (1 δ)100 < x < (1 δ)300, there is a unique equilibrium: hb, Li. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
25 Equilibrium Under Ambiguity (EUA) Theorem The game has the following Equilibria under Ambiguity: 1 When 0 x (1 δ)75, there are 3 equilibria: ht, Mi, hb, Li and 1 4 T B, 1 4 L M. 2 When (1 δ)75 < xd (1 δ)100, there are 3 equilibria: ht, Mi, hb, Li and x (1 δ)100 T + [(1 δ)100 x ] (1 δ)100 B, 1 61 M R E. 3 When (1 δ)100 < x < (1 δ)300, there is a unique equilibrium: hb, Li. 4 When x > (1 δ)300, there is a unique equilibrium: hb, Ri. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
26 Equilibrium Under Ambiguity (EUA) Theorem The game has the following Equilibria under Ambiguity: 1 When 0 x (1 δ)75, there are 3 equilibria: ht, Mi, hb, Li and 1 4 T B, 1 4 L M. 2 When (1 δ)75 < xd (1 δ)100, there are 3 equilibria: ht, Mi, hb, Li and x (1 δ)100 T + [(1 δ)100 x ] (1 δ)100 B, 1 61 M R E. 3 When (1 δ)100 < x < (1 δ)300, there is a unique equilibrium: hb, Li. 4 When x > (1 δ)300, there is a unique equilibrium: hb, Ri. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
27 Equilibrium Under Ambiguity (EUA) Theorem The game has the following Equilibria under Ambiguity: 1 When 0 x (1 δ)75, there are 3 equilibria: ht, Mi, hb, Li and 1 4 T B, 1 4 L M. 2 When (1 δ)75 < xd (1 δ)100, there are 3 equilibria: ht, Mi, hb, Li and x (1 δ)100 T + [(1 δ)100 x ] (1 δ)100 B, 1 61 M R E. 3 When (1 δ)100 < x < (1 δ)300, there is a unique equilibrium: hb, Li. 4 When x > (1 δ)300, there is a unique equilibrium: hb, Ri. If δ = 1 2, then (1) occurs for 0 6 x (2) occurs for x 6 50, (3) occurs for 50 6 x (4) occurs for x. The estimate δ = 1 2, comes from Kilka and Weber David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
28 Experimental Predictions Player 1 x = 60, 120, 170, 200, 230, 260. Player 2 L M R T 0, 0 300, , x B 100, 300 0, 0 55, x By choosing R player 1 is able to protect himself/herself against ambiguity. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
29 Experimental Predictions Player 1 x = 60, 120, 170, 200, 230, 260. Player 2 L M R T 0, 0 300, , x B 100, 300 0, 0 55, x By choosing R player 1 is able to protect himself/herself against ambiguity. If we use δ = 1 2, then R is part of an EUA for x = 60, 170, 200, 230, 260. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
30 Experimental Predictions Player 1 x = 60, 120, 170, 200, 230, 260. Player 2 L M R T 0, 0 300, , x B 100, 300 0, 0 55, x By choosing R player 1 is able to protect himself/herself against ambiguity. If we use δ = 1 2, then R is part of an EUA for x = 60, 170, 200, 230, 260. Nash equilibrium predicts R is not chosen for any of these values of x. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
31 Experimental Predictions Player 1 x = 60, 120, 170, 200, 230, 260. Player 2 L M R T 0, 0 300, , x B 100, 300 0, 0 55, x By choosing R player 1 is able to protect himself/herself against ambiguity. If we use δ = 1 2, then R is part of an EUA for x = 60, 170, 200, 230, 260. Nash equilibrium predicts R is not chosen for any of these values of x. Iterated dominance predicts R is not chosen for x = 120, 170, 200, 230, 260. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
32 Experimental Predictions Player 1 x = 60, 120, 170, 200, 230, 260. Player 2 L M R T 0, 0 300, , x B 100, 300 0, 0 55, x By choosing R player 1 is able to protect himself/herself against ambiguity. If we use δ = 1 2, then R is part of an EUA for x = 60, 170, 200, 230, 260. Nash equilibrium predicts R is not chosen for any of these values of x. Iterated dominance predicts R is not chosen for x = 120, 170, 200, 230, 260. For x = 60, R is dominated by the mixed strategy 3 4 M L which yields an expected return of 75. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
33 Urn Experiments We performed some experiments based on the 3-ball Ellsberg urn. 30 balls 60 balls Act Red Blue Yellow f y 0 0 g h For y = 95, 90, 80, subjects were given the choice of betting on blue or yellow. For y = 100, subjects were given two choices as in the classic Ellsberg experiment. A computer simulated the draw of the ball. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
34 Experimental Design 80 subjects. from St. Stephen s College, Delhi, and University of Exeter. Paper and pencil. Subjects were randomly allocated role of Column Player/Row Player. 11 rounds - Battle of Sexes alternated with an Ellsberg Urn problem. Once choices were made: Random game and Random Ellsberg Urn round chosen to determine payment. Average payment: Indian subjects = Rs.420 ( 6 approx.) Exeter subjects = David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
35 Column Player David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
36 Results: Games There is strong evidence that ambiguity-aversion a ects choice. Strategy R is chosen very frequently. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
37 Results: Games There is strong evidence that ambiguity-aversion a ects choice. Strategy R is chosen very frequently. In cases x = 120, 170, 200, 230, 260, strategy R is chosen even though both Nash equilibrium and iterated dominance say that it should not be. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
38 Results: Games There is strong evidence that ambiguity-aversion a ects choice. Strategy R is chosen very frequently. In cases x = 120, 170, 200, 230, 260, strategy R is chosen even though both Nash equilibrium and iterated dominance say that it should not be. In the case x = 60, 30% of subjects choose R, even though it is dominated by a mixed strategy. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
39 Results: Games There is strong evidence that ambiguity-aversion a ects choice. Strategy R is chosen very frequently. In cases x = 120, 170, 200, 230, 260, strategy R is chosen even though both Nash equilibrium and iterated dominance say that it should not be. In the case x = 60, 30% of subjects choose R, even though it is dominated by a mixed strategy. Row players appear to randomise 50:50. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
40 Results: Games There is strong evidence that ambiguity-aversion a ects choice. Strategy R is chosen very frequently. In cases x = 120, 170, 200, 230, 260, strategy R is chosen even though both Nash equilibrium and iterated dominance say that it should not be. In the case x = 60, 30% of subjects choose R, even though it is dominated by a mixed strategy. Row players appear to randomise 50:50. There was no signi cant e ect of: David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
41 Results: Games There is strong evidence that ambiguity-aversion a ects choice. Strategy R is chosen very frequently. In cases x = 120, 170, 200, 230, 260, strategy R is chosen even though both Nash equilibrium and iterated dominance say that it should not be. In the case x = 60, 30% of subjects choose R, even though it is dominated by a mixed strategy. Row players appear to randomise 50:50. There was no signi cant e ect of: Gender, David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
42 Results: Games There is strong evidence that ambiguity-aversion a ects choice. Strategy R is chosen very frequently. In cases x = 120, 170, 200, 230, 260, strategy R is chosen even though both Nash equilibrium and iterated dominance say that it should not be. In the case x = 60, 30% of subjects choose R, even though it is dominated by a mixed strategy. Row players appear to randomise 50:50. There was no signi cant e ect of: Gender, Location, i.e. Exeter or Delhi, David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
43 Results: Games There is strong evidence that ambiguity-aversion a ects choice. Strategy R is chosen very frequently. In cases x = 120, 170, 200, 230, 260, strategy R is chosen even though both Nash equilibrium and iterated dominance say that it should not be. In the case x = 60, 30% of subjects choose R, even though it is dominated by a mixed strategy. Row players appear to randomise 50:50. There was no signi cant e ect of: Gender, Location, i.e. Exeter or Delhi, degree programme, i.e. arts or science. (Economics students were not used as subjects.) David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
44 David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
45 David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
46 Results of Urn Experiments When y = 100 : David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
47 Results of Urn Experiments When y = 100 : 46% of subjects displayed ambiguity-aversion. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
48 Results of Urn Experiments When y = 100 : 46% of subjects displayed ambiguity-aversion. 9% of subjects displayed ambiguity-preference. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
49 Results of Urn Experiments When y = 100 : 46% of subjects displayed ambiguity-aversion. 9% of subjects displayed ambiguity-preference. Less ambiguity-aversion than in most previous experiments. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
50 Results of Urn Experiments When y = 100 : 46% of subjects displayed ambiguity-aversion. 9% of subjects displayed ambiguity-preference. Less ambiguity-aversion than in most previous experiments. Due to computer simulation? David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
51 Results of Urn Experiments When y = 100 : 46% of subjects displayed ambiguity-aversion. 9% of subjects displayed ambiguity-preference. Less ambiguity-aversion than in most previous experiments. Due to computer simulation? David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
52 Results of Urn Experiments When y = 100 : 46% of subjects displayed ambiguity-aversion. 9% of subjects displayed ambiguity-preference. Less ambiguity-aversion than in most previous experiments. Due to computer simulation? Less ambiguity-aversion than in the game experiments. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
53 Results of Urn Experiments When y = 100 : 46% of subjects displayed ambiguity-aversion. 9% of subjects displayed ambiguity-preference. Less ambiguity-aversion than in most previous experiments. Due to computer simulation? Less ambiguity-aversion than in the game experiments. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
54 Results of Urn Experiments When y = 100 : 46% of subjects displayed ambiguity-aversion. 9% of subjects displayed ambiguity-preference. Less ambiguity-aversion than in most previous experiments. Due to computer simulation? Less ambiguity-aversion than in the game experiments. When y = 80, 30% of subjects chose R, thus displaying strong ambiguity-aversion. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
55 Results of Urn Experiments When y = 100 : 46% of subjects displayed ambiguity-aversion. 9% of subjects displayed ambiguity-preference. Less ambiguity-aversion than in most previous experiments. Due to computer simulation? Less ambiguity-aversion than in the game experiments. When y = 80, 30% of subjects chose R, thus displaying strong ambiguity-aversion. Correlation between ambiguity-attitude in games and urn experiments is low. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
56 Conclusion Ambiguity a ects play in games. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
57 Conclusion Ambiguity a ects play in games. There is more ambiguity-aversion in games than in urn experiments. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
58 Conclusion Ambiguity a ects play in games. There is more ambiguity-aversion in games than in urn experiments. Subjects perceive the urn as less ambiguous. For instance they can assign probabilities by the principal of insu cient reason. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
59 Conclusion Ambiguity a ects play in games. There is more ambiguity-aversion in games than in urn experiments. Subjects perceive the urn as less ambiguous. For instance they can assign probabilities by the principal of insu cient reason. Less ambiguity aversion than in previous Ellsberg urn experiments. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
60 Conclusion Ambiguity a ects play in games. There is more ambiguity-aversion in games than in urn experiments. Subjects perceive the urn as less ambiguous. For instance they can assign probabilities by the principal of insu cient reason. Less ambiguity aversion than in previous Ellsberg urn experiments. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
61 Conclusion Ambiguity a ects play in games. There is more ambiguity-aversion in games than in urn experiments. Subjects perceive the urn as less ambiguous. For instance they can assign probabilities by the principal of insu cient reason. Less ambiguity aversion than in previous Ellsberg urn experiments. Lo (1996) presents an alternative theory of ambiguity in games. For games with only pure strategy Nash equilibrium Lo s solution concept coincides with Nash equilibrium. Dow and Werlang (1994) theory of ambiguity in games provides a better explanation of our evidence than Lo s theory. David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
62 Chateauneuf, A., J. Eichberger, and S. Grant (2007): Choice under Uncertainty with the Best and Worst in Mind: NEO-Additive Capacities, Journal of Economic Theory, 137, Dow, J., and S. R. C. Werlang (1994): Nash Equilibrium under Uncertainty: Breaking Down Backward Induction, Journal of Economic Theory, 64, Eichberger, J., D. Kelsey, and B. Schipper (2009): Ambiguity and Social Interaction, Oxford Economic Papers, 61, Lo, K. C. (1996): Equilibrium in Beliefs under Uncertainty, Journal of Economic Theory, 71, Schmeidler, D. (1989): Subjective Probability and Expected Utility without Additivity, Econometrica, 57, David Kelsey (University of Exeter.) Ambiguity in Coordination Games September / 17
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