Session 2: Experimental and Behavioral Game Theory David Reiley Yahoo! Research January 2011

Experimental economists test the predictions of game theory. Morning session: laboratory experiments. Afternoon session: field experiments.

Experimental economists test the predictions of game theory. Morning session: laboratory experiments. Afternoon session: field experiments. How many of you have been a subject in a game-theory experiment?

Let’s play the Travelers’ Dilemma. Two players simultaneously choose a “claim” between 180 and 300 (cents). Both players receive an amount equal to the lower of the two claims. The player who submits the lower claim gets an additional reward of R=180. The player who submits the higher claim loses a penalty of R=180.

Now let’s play another version. Two players simultaneously choose a “claim” between 180 and 300 (cents). Both players receive an amount equal to the lower of the two claims. The player who submits the lower claim gets an additional reward of R=5. The player who submits the higher claim loses a penalty of R=5.

Exercise: What are the rationalizable strategies for this game with R=5?

Answer: x=180 is the only rationalizable strategy. Exercise: What are the rationalizable strategies for the game with R=5? Answer: x=180 is the only rationalizable strategy. If you play y, my best response is x=y-1. So x=300 is never a best response. If I know you will never play y=300, then x=299 is never a best response. … Iterated elimination of these dominated strategies continues until we are left only with x=180.

What are the Nash equilibrium predictions for the two games?

What are the Nash equilibrium predictions for the two games? Symmetric equilibrium with x=180. Same equilibrium both for R=5 and for R=180.

How does the QRE prediction vary across games? The predicted distribution of strategies depends on the error parameter µ. But comparative statics are clear: Claims should be higher when the penalty/reward R is lower (R=5). The cost to making a mistake is lower when R=5. Knowing this about my opponent, I best-respond by making higher claims.

What do the data look like? Light bars for R=180, dark bars for R=5.

See Holt’s spreadsheet for an interactive, graphical illustration of QRE in this game.

Some principles and guidelines that improve my (game-theory) teaching. I reserve the scarce resource of class time for two-way communication as much as possible. The book can’t answer questions, but I can. One-way communication comes from the reading. Professor as coach. Students must earn their learning. My assigned practice regime should help them earn it as efficiently as possible. Graded online quizzes promote reading before class. Short-term incentives help students manage their learning. Playing games gets students to understand the strategies, so that they can appreciate the theory. $30 lab fee at beginning of course.

Is game theory a science, or a philosophy? Explain your opinion. Goeree and Holt: “Ten Little Treasures and Ten Intuitive Contradictions.” Is game theory a science, or a philosophy? Explain your opinion. Does Selten characterize game theory as positive or normative? What did Nash conclude from his own game-theory experiments?

What are the advantages and disadvantages of studying one-shot games in the laboratory?

Disadvantage: Advantages: What are the advantages and disadvantages of studying one-shot games in the laboratory? Disadvantage: When subjects play an unfamiliar game, they may not play as they would in a real-world setting where they have experience. Advantages: Don’t have to worry about supergames. Games are independent. Many real-world games involve one-shot interactions and little experience.

Now let’s play the asymmetric matching-pennies game from Table 1. Left Right Top 320, 40 40, 80 Bottom 80, 40

What is the Nash equilibrium of this game? Left Right Top 320, 40 40, 80 Bottom 80, 40

What is the Nash equilibrium of this game? Left Right Top 320, 40 40, 80 Bottom 80, 40 Rowena: (1/2) Top + (1/2) Bottom Colin: (1/8) Left + (7/8) Right

What is the Nash equilibrium of this second asymmetric MP game? Left Right Top 44, 40 40, 80 Bottom 80, 40 Rowena: (1/2)Top + (1/2)Bottom Colin: (10/11)Left + (1/11)Right

How does the Nash equilibrium shift from the symmetric to the asymmetric Matching Pennies game?

How does the Nash equilibrium shift from the symmetric to the asymmetric Matching Pennies game? Row player’s equilibrium strategy doesn’t change (p=1/2) when her payoffs change. Column player’s equilibrium strategy changes (q=1/2 to q=1/8), even though his payoffs don’t change.

What happens in the actual data on Matching Pennies?

What happens in the actual data on Matching Pennies? Row player’s choices change dramatically, even though the Nash equilibrium prediction doesn’t change. Can QRE explain this? (Spreadsheet.)

QRE has done well without overfitting. Estimating the error parameter (µ) on one game, and applying it to an entirely different game, yields impressively good predictions in many cases. Very different from estimating a separate µ on the data from each separate game.

Next, let’s play a minimum-effort coordination game. Everyone will play in one group. Each player chooses an effort level ei from {110,111,112,…,170}. The group output per person equals the minimum of the efforts. Each individual pays 0.1x her effort. So total payoff to player i is: πi = min(ej)-0.1ei

How many Nash equilibria are there to this game?

How many Nash equilibria are there to this game? 61 different symmetric equilibria. Any symmetric choice is an equilibrium. Why is there no gain to deviating? Why are asymmetric pure strategies not an equilibrium? The set of Nash equilibria doesn’t depend on c (or N.)

How could we choose among the multiple equilibria? Payoff dominance? Risk dominance? Note that this is defined only for 2x2 games. What happens in practice?

Results from the minimum-effort coordination game. Again, data show comparative statics not predicted by Nash equilibrium.

What is the main message of this paper? Goeree and Holt: “Ten Little Treasures and Ten Intuitive Contradictions.” What is the main message of this paper? What are the treasures, and what are the contradictions?

Behavioral game theory (BGT) seeks to incorporate psychology to improve the theory’s predictive power. Today’s reading assignment identifies three main assumptions that BGT has relaxed relative to rational GT. What are they?

Perfect decisionmaking Perfect foresight (correct beliefs) Behavioral game theory (BGT) seeks to incorporate psychology to improve the theory’s predictive power. Today’s reading assignment identifies three main assumptions that BGT has relaxed relative to rational GT. What are they? Perfect selfishness Perfect decisionmaking Perfect foresight (correct beliefs)

Two kinds of relaxations to the assumption of selfishness: Inequity aversion True payoffs are a function of the monetary payoffs, reflecting dislike for inequality Fehr/Schmidt, Bolton/Ockenfels Psychological game theory Payoffs depend on the beliefs and expectations about the other player Reciprocal fairness (Rabin) Guilt aversion (Charness/Dufwenberg)

Level-k reasoning models Relaxations to the assumption of perfect foresight - see Crawford, Costa-Gomes, and Iriberri (2010). k-rationalizability (k iterations of elimination of dominated strategies) Includes Nash and non-Nash outcomes Level-k reasoning models Type 0 plays uniformly over strategies Type 1 best-responds to type 0 Type n best-responds to type n-1 Noisy introspection (Goeree and Holt)

QRE involves errors that are more likely the less costly they are. Quantal Response Equilibrium (QRE) relaxes the assumption of perfect decisionmaking. Early statistical analyses of experiments included an i.i.d. error term additive to the predicted strategy. QRE involves errors that are more likely the less costly they are. This model is sophisticated about beliefs. Players correctly understand the distribution of other players’ errors, and best-respond to them.

Let’s play a guessing-game called the “p-beauty contest.” Entire group plays together. Each player writes down a number from 0 to 100, inclusive. I will compute the average (X) of the numbers. The person closest to (2/3)X wins $20. I will randomize if there are ties.

Keynes on the original beauty-contest game. “...professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one’s judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practice the fourth, fifth and higher degrees.”

Data from an experiment run in The Financial Times by Richard Thaler, reported in Bosch-Domenesch et al. Note the spikes at 22 and 33. These are consistent with level-k reasoning.