Correlated equilibria, good and bad: an experimental study John Duffy (University of Pittsburgh) Nick Feltovich (University of Aberdeen) n.feltovich@abdn.ac.uk The preliminary paper can be found at www.abdn.ac.uk/~pec214/papers/correl.pdf

Background Player 2 D C Player 1 0,0 9,3 3,9 7,7 Consider this version of the Chicken game: Nash equilibria are (D,C), (C,D), and a mixed equilibrium with Prob(C)=3/5. The mixed NE is symmetric, but yields low payoffs (5.4 for each player). Are there equitable outcomes where the players do better?

Background Player 2 D C Player 1 0,0 9,3 3,9 7,7 Aumann (1974): If the players can observe a public coin toss, they can get payoffs of 6 each, instead of the mixed NE payoff of 5.4 each. E.g. Player 1 chooses C after Heads, D after Tails; Player 2 chooses D after Heads, C after Tails. In this case (unlike mixed-strategy Nash equilibrium), players’ strategies are not statistically independent. Correlated equilibrium: generalisation of Nash equilibrium that allows strategies to be correlated across the players.

Background Player 2 D C Player 1 0,0 9,3 3,9 7,7 Aumann (1974): If there is a third party (non-strategic player), the players can do better still. Suppose the third party chooses the outcomes (D,C), (C,D), and (C,C) with equal probability, and “recommends” the corresponding action to each player (but not the opponent)—and suppose the third party’s behaviour is common knowledge between Players 1 and 2. Then, if the players follow their recommendations, expected payoffs are 6-1/3 each.

Background Player 2 D C Player 1 0,0 9,3 3,9 7,7 Not only does following these recommendations raise players’ payoffs, it is equilibrium behaviour! Why? Suppose Player 1 receives a D recommendation. Then she knows for sure that Player 2’s recommendation is C. If she expects Player 2 to choose C, then U1(D)=9>7=U1(C). Suppose Player 1 receives a C recommendation. Then she knows that Player 2 got either a C or a D recommendation with probability 0.5. If she expects Player 2 to follow his recommendation, then and , so U1(C)>U1(D).

Background Correlated equilibrium allows for higher symmetric payoffs than in any Nash equilibrium—but also for lower symmetric payoffs. 9 6 Player 2 payoff 3 3 6 9 Player 1 payoff

Research questions (a) Is it possible to implement correlated equilibria when actual people are playing this game? (I.e., do people follow recommendations?) (b) Does the answer to (a) depend on which outcome distribution is chosen? We consider two correlated non-Nash equilibria: “Good recommendations”: “Bad recommendations”: Player 2 D C Player 1 1/3 Player 2 D C Player 1 1/5 2/5 Expected payoff: ≈6.333 Expected payoff: 4.8

Research questions We also consider: “Nash recommendations” (correlated equilibrium that is a convex combination of Nash equilibria): “Very good recommendations” (not a correlated equilibrium): Player 2 D C Player 1 1/2 Player 2 D C Player 1 1/10 4/5 Expected payoff: 6 Expected payoff: 6.8

Previous research Cason and Sharma (2007 Economic Theory): Similar game, correlated equilibrium with good recommendations. Player 2 Left Right Player 1 Up 3,3 (0.000) 48,9 (0.375) Down 9,48 (0.375) 39,39 (0.250) Results: Subjects followed Up/Left (D) recommendations roughly 85% of the time, Down/Right (C) recommendations roughly 75% of the time. Adding recommendations to the game raised average payoffs by about 15% (vs. predicted 35%).

Experimental design Experimental sessions took place at Pittsburgh Experimental Economics Laboratory (PEEL), with all interaction via networked computers. Subjects were randomly matched in each round (12 subjects in each session). All subjects played 20 rounds without recommendations, 20 rounds with recommendations (within-subject variation). Order of games (with recommendations or without)—varied across subjects. Good, bad, Nash, or very good recommendations—varied across subjects.

Experimental design Instructions—including the outcome distribution underlying the recommendations—were presented orally (as well as in writing), in an attempt to satisfy common knowledge of the situation. Actions labelled as “X” (=D) and “Y” (=C). Feedback: own and opponent recommendation, action, payoff at the end of each round Payments: $5 show-up fee plus $1/point earned in two randomly chosen rounds (one from rounds 1-20, one from rounds 21-40). Average payment of roughly $15 for a 45-60 minute session. Recommendations explained in neutral language.

Experimental design Excerpt from our instructions (good-recommendations treatment): “Recommendations: Before choosing an action in a round, both you and the person you’re matched with are given recommendations by the computer program. Different recommendations will be given in different rounds. In any round, there are three possibilities: There is a ⅓ (33⅓%) chance that it will be recommended that you choose [D] and the other player choose [C]; There is a ⅓ (33⅓%) chance that it will be recommended that you choose [C] and the other player choose [D]; There is a ⅓ (33⅓%) chance that it will be recommended that both you and the other player choose [C]; It will never happen that you are both recommended to choose [D]. These recommendations are optional; it is up to you whether or not to follow them. [Emphasis added.] Notice that your recommendation may give you information about the recommendation that was given to the person matched to you.”

Comparison with other procedures Excerpt from Cason/Sharma instructions: “Why you should follow the recommendations You should follow the recommendation given by the computer, because as long as the person you are paired with also follows his or her recommendation then you earn more on average by following the recommendation. Here is why: 1. First, remember that if both you and the participant you are paired with follow the recommendations, you will never have the worst Up-Left outcome (in which both participants earn only 3), because that outcome is never recommended. 2. Next, if you are a Red participant and you receive the recommendation to choose Up, then you know that the Blue participant you are paired with has received the recommendation to choose Right, since the outcome Up-Left is never recommended. [You know that a green or red ball was not drawn, since they recommend Down.] If this Blue participant follows his recommendation and chooses Right, then you earn more by following your recommendation to choose Up (48) than by not following your recommendation and choosing Down (39). […] To reiterate: you always earn more by following your recommendation as long as the participant you are paired with also follows his or her recommendation.”

Experimental results Aggregate outcome frequencies [implied from recommendations]: (C,C) (C,D) or (D,C) (D,D) Mean payoff No recommendations .347 .494 .159 5.393 Good recommendations .281 [.333] .579 [.667] .140 [.000] 5.444 [6.333] Bad recommendations .327 .481 [.800] .192 [.200] 4.902 [4.800] Nash recommendations .323 .565 [1.000] .112 5.648 [6.000] Very good recommendations .325 .466 .208 5.075 [6.800] Mixed NE prediction .360 .480 .160 5.400

Experimental results Frequency of followed Frequency of followed

Experimental results Parametric statistics—methodology Probit models Dependent variable: indicator for C choice (1=yes, 0=no) Independent variables: — round number; indicator for recommendations-first ordering — indicator for C recommendation, product with round number — indicator for D recommendation, product with round number Individual-subject random effects, some models with session fixed effects Separate estimations for each treatment STATA (v. 10) We estimate the incremental effect of a C recommendation on the likelihood of a C choice in round t: Φ(X∙B + βCrec + βCrec∙round ∙ t) − Φ(X∙B) (analogous formula for D recommendation)

Experimental results Estimated incremental effect of recommendations on C choice

Experimental results Are individual subjects better off if they follow recommendations? To answer this, we consider foregone payoffs: (Payoff from choosing other action) − (Payoff actually earned) => Do subjects who follow recommendations more often have lower foregone payoffs?

Experimental results Association between following recommendations and foregone payoffs (individual subjects, 20-round averages): Good, Nash recommendations Bad, very good recommendations

Summary Giving recommendations to subjects has an effect on the distribution of outcomes. Recommendations are followed more often than chance (mixed-strategy Nash equilibrium) would predict, but far less than 100%. The frequency of followed recommendations varies by treatment—they are more likely to be followed when they come from a correlated equilibrium; they raise payoffs.

Next steps Next paper: “endogenous correlated equilibrium”—will a strategic third party make recommendations that form a correlated equilibrium? If so, what kind of correlated equilibrium?