Epistemic Conditions for the Failure of Nash Equilibrium
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1 Epistemic Conditions for the Failure of Nash Equilibrium P.J. Healy SITE August PJ Healy (OSU) Epistemics / 48
2 PJ Healy (OSU) Epistemics / 48
3 Motivation Question: Do people play Nash equilibrium? If not, why not? Utilities? (NE with Fehr-Schmidt preferences) Beliefs? (Level-K) Rationality? (QRE) Rationality is not primitive. Imposed by the solution concept?... not so obvious. We need theory framework in which to discuss these things! PJ Healy (OSU) Epistemics 201 / 48
4 Aumann & Brandenburger (15), bastardized The lab environment: Outcome space: X ($$ payoffs to each player) Game form: (N, S, g) (g : S X) PJ Healy (OSU) Epistemics / 48
5 Example: 1-Shot P.D. in the Lab L R U 2, 2 0, D, 0 1, 1 A Game L R U $2,$2 $0,$ D $,$0 $1,$1 A Game Form ւ ց L R U 4, 4 0, 0 D 0, 0 2, 2 A Game L R U 0, 0 0, 0 D 0, 0 0, 0 A Game Actual game is determined by players types. PJ Healy (OSU) Epistemics / 48
6 Adding Epistemology Outcome space: X Game form: (N, S, g) (g : S X) Type space: Θ i for each i N Player s type θ i determines: Payoff function: u i (x; θ i ) Beliefs about θ i : p i (θ i )(θ i ) Pure strategy choice: s i (θ i ) S i Cool! Given this, we can define at each θ i : i s conjecture about s i : φ i (θ i )(s i ) = p i (θ i )({θ i : s i (θ i ) = s i }) i s (subjective) expected utility: Eu i (s i ; θ i ) = s i [φ i (θ i )(s i )] u i (g(s i, s i ); θ i ) PJ Healy (OSU) Epistemics 201 / 48
7 Rationality & Equilibrium A player is rational at θ i if: s i (θ i ) arg max s i Eu i (s i ; θ i ). Standard definitions of known, mutually known, and common knowledge. PJ Healy (OSU) Epistemics / 48
8 Interpretations Everyone comes to the lab with a θ i Preferences over outcomes (inequality aversion, selfishness, etc.) captured in u i (x; θ i ) Nobody mixes: I m uncertain about your action only because I m uncertain about your type. Thus, mixed-strategy equilibrium only exists in conjectures. Equilibrium is a property of beliefs, not actions! PJ Healy (OSU) Epistemics / 48
9 AB5 s Theorem: 2 Players Theorem Suppose n = 2. If 1 u(θ) is mutually known, 2 φ(θ) is mutually known, and rationality is mutually known. Then (φ 2 (θ 2 ), φ 1 (θ 1 )) is a MSNE of (N, S, u g). PJ Healy (OSU) Epistemics 201 / 48
10 This Paper Subjects play five 2 2 one-shot games. Strangers, no feedback. For each game, elicit: 1 Chosen action (s i (θ i )) Play game 2 Preferences over outcomes (u i ( ; θ i )) Probability equivalent of each cell. u(x) [0, 100] i s beliefs about u j Point estimate, paid on abs. deviation 4 Conjectures about s j (φ i (θ i )) Grether/Karni mechanism (probability BDM) 5 i s beliefs about φ j. Point estimate, paid on abs. deviation i s beliefs about j s rationality Grether/Karni mechanism If (φ 2, φ 1 ) is not NE, then 1 of these conditions fails. WHICH?? PJ Healy (OSU) Epistemics / 48
11 Example Observation 0% 100% >0% 50, 50 0, 0 100% 10, 10 1, 1 L R U $ 10, 5 $ 15, 15 D $ 5, 10 $ 1, 1 Obs 180: Row s Game Game Form 55% 45% 85% 40, 10 50, 70 15% 10, 20 5, 5 Obs 180: Column s Game 0% 100% >0%, 4, 4 100% 2, 2 1, 1 Row s Ordinal Game 55% 45% 85%, 2 4, 4 15% 2, 1, 1 Column s Ordinal Game PJ Healy (OSU) Epistemics / 48
12 Data 2010 Data: 78 subjects Very negative results. Confusing interface? (Note: blame RA) 2011 Data: More intuitive interface & instructions 72 subjects 201 Data: Simple pencil & paper Ordinal preferences, guess s j, no rationality. Can t test AB subjects so far 0 min, $5 $20 payout PJ Healy (OSU) Epistemics 201 / 48
13 Game Form 1: Dominance Solvable Game Form: L R U $ 10, 5 $ 15, 15 D $ 5, 10 $ 1, 1 Dom. Solvable $NE: (U,R) PJ Healy (OSU) Epistemics / 48
14 Game Form 2: Symmetric Coordination Game Form: L R U $ 15, 15 $ 1, 1 D $ 1, 1 $ 5, 5 Three $NE: (U,L) (D,R) ((2/,7/),(2/,7/)) PJ Healy (OSU) Epistemics / 48
15 Game Form : Prisoners Dilemma Game Form: L R U $ 10, 10 $ 1, 15 D $ 15, 1 $ 5, 5 Dominant Strategy Equil ($): (D,R) PJ Healy (OSU) Epistemics / 48
16 Game Form 4: Asymmetric Matching Pennies Game Form: L R U $ 15, 5 $ 5, 10 D $ 5, 10 $ 10, 5 Unique Mixed-Strategy $NE: ((1/2,1/2),(1/,2/)) PJ Healy (OSU) Epistemics / 48
17 Game Form 5: Asymmetric Coordination Game Form: L R U $ 15, 5 $ 1, 1 D $ 1, 1 $ 5, 10 Three $NE: (U,L) (D,R) ((/1,4/1),(2/,7/)) PJ Healy (OSU) Epistemics / 48
18 Hand-Waving Summary of 2011 Data Game Form $ 10, $ 5 $ 15, $ 15 G1:DomSolv $ 5, $ 10 $ 1, $ 1 G2:SymCoord G:PD G4:AsymMP G5:AsymCoord $ 15, $ 15 $ 1, $ 1 $ 1, $ 1 $ 5, $ 5 $ 10, $ 10 $ 1, $ 15 $ 15, $ 1 $ 5, $ 5 $ 15, $ 5 $ 5, $ 10 $ 5, $ 10 $ 10, $ 5 $ 15, $ 5 $ 1, $ 1 $ 1, $ 1 $ 5, $ 10 Util: Decent Blfs: Bad Util: Blfs: Util: V.Bad Blfs: Bad Util: Weak Blfs: Util: OK Blfs: Bad RowRtnl: ColRtnl: OK RowRtnl: ColRtnl: RowRtnl: OK ColRtnl: OK RowRtnl: Bad ColRtnl: Weak RowRtnl: Bad ColRtnl: Bad PJ Healy (OSU) Epistemics / 48
19 Playing the Same Ordinal Game? Game Form $ 10, $ 5 $ 15, $ 15 G1:DomSolv $ 5, $ 10 $ 1, $ 1 Row=Col =GameForm.4% 100% G2:SymCoord G:PD G4:AsymMP $ 15, $ 15 $ 1, $ 1 $ 1, $ 1 $ 5, $ 5 $ 10, $ 10 $ 1, $ 15 $ 15, $ 1 $ 5, $ 5 $ 15, $ 5 $ 5, $ 10 $ 5, $ 10 $ 10, $ 5 88.% 100%.1% 2.% 52.8% 100% G5:AsymCoord $ 15, $ 5 $ 1, $ 1 $ 1, $ 1 $ 5, $ % 100% Overall: 4.4%.1% H O : Random Response:.25%.25% PJ Healy (OSU) Epistemics / 48
20 Testing AB5 w/ 2011 Data u mutual knowledge: If same ordinal game φ mutual knowledge: If max ±10% error Ratn l mutual knowledge: If true, >= 75% prob 10/180 observations satisfy these conditions. : Game 2 (SymCoord). 1: Game 1 (DomSolv) PJ Healy (OSU) Epistemics / 48
21 The 201 Data Simple enough that I trust the data Not rich enough to test at the individual (pair) level PJ Healy (OSU) Epistemics / 48
22 201 Data: Preference Ranking Histograms GAME 1: DOMINANCE SOLVABLE L R U $ 10, 5 $ 15, 15 D $ 5, 10 $ 1, 1 Row Prefs Col Prefs PJ Healy (OSU) Epistemics / 48
23 Play, Conjectures & Beliefs) GAME 1: DOMINANCE SOLVABLE Play 15% 85% 1stBlf 8% 2% Play 1stBlf 2ndBlf 15% 85% 100% 2% 100% $10,$5 $15,$15 0% 8% 0% $5,$10 $1,$1 Utility Beliefs OK Rationality seems to fail, but maybe not at indiv. level PJ Healy (OSU) Epistemics / 48
24 201 Data: Preference Ranking Histograms GAME 2: SYMMETRIC COORDINATION L R U $ 15, 15 $ 1, 1 D $ 1, 1 $ 5, 5 Row Prefs Col Prefs PJ Healy (OSU) Epistemics / 48
25 Play, Conjectures & Beliefs) GAME 2: SYMMETRIC COORDINATION Play 2% 8% 1stBlf 100% 0% Play 1stBlf 2ndBlf 100% 0% 100% 100% 100% $15,$15 $1,$1 0% 0% 0% $1,$1 $5,$5 Utility Beliefs A game theory home run!! PJ Healy (OSU) Epistemics / 48
26 201 Data: Preference Ranking Histograms GAME : PRISONERS DILEMMA L R U $ 10, 10 $ 1, 15 D $ 15, 1 $ 5, 5 Row Prefs Col Prefs PJ Healy (OSU) Epistemics / 48
27 Play, Conjectures & Beliefs) GAME : PRISONERS DILEMMA Play 54% 4% 1stBlf 1% % Play 1stBlf 2ndBlf 4% 54% 54% 1% 8% $10,$10 $1,$15 4% % 2% $15,$1 $5,$5 Utility X Beliefs X Not complete-info game. PJ Healy (OSU) Epistemics / 48
28 201 Data: Preference Ranking Histograms GAME 4: ASYMMETRIC MATCHING PENNIES L R U $ 15, 5 $ 5, 10 D $ 5, 10 $ 10, 5 Row Prefs Col Prefs PJ Healy (OSU) Epistemics / 48
29 Play, Conjectures & Beliefs) GAME 4: ASYMMETRIC MATCHING PENNIES Play 54% 4% 1stBlf % 1% Play 1stBlf 2ndBlf 2% 8% 85% 2% 100% $15,$5 $5,$10 15% 8% 0% $5,$10 $10,$5 Utility OK, but a little shaky Beliefs Rationality seems to fail for Column PJ Healy (OSU) Epistemics / 48
30 201 Data: Preference Ranking Histograms GAME 5: ASYMMETRIC COORDINATION L R U $ 15, 5 $ 1, 1 D $ 1, 1 $ 5, 10 Row Prefs Col Prefs PJ Healy (OSU) Epistemics / 48
31 Play, Conjectures & Beliefs) GAME 5: ASYMMETRIC COORDINATION Play 54% 4% 1stBlf 1% % Play 1stBlf 2ndBlf 2% 77% 2% 100% 2% $15,$5 $1,$1 8% 0% 8% $1,$1 $5,$10 Utility Beliefs OK Rationality seems to fail for Column PJ Healy (OSU) Epistemics / 48
32 Summary Sometimes not even playing same game! NE not defined Subjects are pretty bad at 2nd order beliefs. Beliefs about rationality are reasonably good. When are utilities mutual knowledge?? Respect for Bayesian games... but beliefs? WARNING: Confound with reliability of elicitation procedure. See: Old data vs. New data PJ Healy (OSU) Epistemics / 48
33 The End. PJ Healy (OSU) Epistemics 201 / 48
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