More Experiments on

Syngjoo Choi

Spring 2010

Experimental Economics (ECON3020) Game theory 2 Spring2010 1/28 Playing Unpredictably

In many situations there is a strategic advantage associated with being unpredictable.

Experimental Economics (ECON3020) Game theory 2 Spring2010 2/28 Playing Unpredictably

In many situations there is a strategic advantage associated with being unpredictable.

Experimental Economics (ECON3020) Game theory 2 Spring2010 2/28 Mixed (or Randomized)

In the penalty kick, Left (L) and Right (R) are the pure strategies of the kicker and the goalkeeper. A mixed strategy refers to a probabilistic mixture over pure strategies in which no single pure strategy is played all the time. e.g., kicking left with half of the time and right with the other half of the time. A penalty kick in a soccer game is one example of games of pure con‡icit, essentially called zero-sum games in which one player’s winning is the other’sloss.

Experimental Economics (ECON3020) Game theory 2 Spring2010 3/28 The use of pure strategies will result in a loss. What about each player playing heads half of the time?

Matching Pennies Games

In a matching pennies game, each player uncovers a penny showing either heads or tails. One player takes both coins if the pennies match; otherwise, the other takes both coins.

Left Right Top 1, 1, 1 1 Bottom 1, 1, 1 1 Does there exist a involving the use of pure strategies?

Experimental Economics (ECON3020) Game theory 2 Spring2010 4/28 Matching Pennies Games

In a matching pennies game, each player uncovers a penny showing either heads or tails. One player takes both coins if the pennies match; otherwise, the other takes both coins.

Left Right Top 1, 1, 1 1 Bottom 1, 1, 1 1 Does there exist a Nash equilibrium involving the use of pure strategies? The use of pure strategies will result in a loss. What about each player playing heads half of the time?

Experimental Economics (ECON3020) Game theory 2 Spring2010 4/28 Mixed-Strategy Equilibrium

In the matching pennies game, both players playing heads with half probability is the unique Nash equilibrium. On one hand, this strategy is used to not be predictable. On the other hand, given the other’sstrategy, there is no (either pure or mixed) strategy giving higher payo¤s than playing head with half probability. One way of seeing this equilibrium is drawing each player’sbest response: Suppose player 2 chooses (Right) with probability p. Then, player 1’spayo¤ di¤erence betwee (Top) and (Bottom) is (1 2p) ( 1 + 2p) = 2 (1 2p). If p< 1/2, it is optimal for player 1 to choose (Top) with probability 1. Similarly for other cases.

Experimental Economics (ECON3020) Game theory 2 Spring2010 5/28 Experimental Economics (ECON3020) Game theory 2 Spring2010 6/28 Classroom Experiment

Now let’splay a game in a classroom experiment. Go to http://veconlab.econ.virginia.edu/login.htm. Session name is sjc8.

Experimental Economics (ECON3020) Game theory 2 Spring2010 7/28 Does there exist a pure-strategy Nash equilibrium? (Intuitive guess) Given the change of payo¤s, who should still play each pure strategy with half probability in an equilibrium? How did you play?

Asymmetric Matching Pennies Game

The game you just played is an asymmetric matching pennies game:

Left Right Top 360, 36, 36 72 Bottom 36, 72, 72 36

Experimental Economics (ECON3020) Game theory 2 Spring2010 8/28 (Intuitive guess) Given the change of payo¤s, who should still play each pure strategy with half probability in an equilibrium? How did you play?

Asymmetric Matching Pennies Game

The game you just played is an asymmetric matching pennies game:

Left Right Top 360, 36, 36 72 Bottom 36, 72, 72 36

Does there exist a pure-strategy Nash equilibrium?

Experimental Economics (ECON3020) Game theory 2 Spring2010 8/28 Asymmetric Matching Pennies Game

The game you just played is an asymmetric matching pennies game:

Left Right Top 360, 36, 36 72 Bottom 36, 72, 72 36

Does there exist a pure-strategy Nash equilibrium? (Intuitive guess) Given the change of payo¤s, who should still play each pure strategy with half probability in an equilibrium? How did you play?

Experimental Economics (ECON3020) Game theory 2 Spring2010 8/28 Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability.

Mixed-strategy Equilibrium

Let’sderive the mixed-strategy equilibrium:

Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability.

Mixed-strategy Equilibrium

Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p.

Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability.

Mixed-strategy Equilibrium

Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p

Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability.

Mixed-strategy Equilibrium

Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p.

Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability.

Mixed-strategy Equilibrium

Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1.

Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability.

Mixed-strategy Equilibrium

Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q.

Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability.

Mixed-strategy Equilibrium

Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q).

Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 Mixed-strategy Equilibrium

Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability.

Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 Experimental Economics (ECON3020) Game theory 2 Spring2010 10/28 Noisy Best Responses

Human subjects’behavior in a laboratory experiment often show noises. One approach of accounting for such noises in experimental data is modeling “probabilistic choice” or “noisy .” Duncan Luce (1959) suggested a way to model noisy choices by assuming that response probabilities are increasing functions of the strength of the stimulus (payo¤s in economics). The probabilistic choice model often uses an exponential function (known as the logit model) in computing choice probabilities of strategies: e.g.,

exp (βπ ) Pr (Right) = right , exp (βπright ) + exp (βπleft )

where πright denotes an expected payo¤ for (Right) and β captures the sensitivity of choice probabilities with respect to payo¤ strength.

Experimental Economics (ECON3020) Game theory 2 Spring2010 11/28 Noisy Best Responses: Symmetric Matching Pennies Game

Experimental Economics (ECON3020) Game theory 2 Spring2010 12/28 Noisy Best Responses: Asymmetric Matching Pennies Game

Experimental Economics (ECON3020) Game theory 2 Spring2010 13/28 Quantal Response Equilibrium I

Standard economic theory relies on the assumption of perfect rationality (especially, with no mistakes in choices). Evaluation of these models using data from either …eld data or laboratory experiments requires an error structure, since choice behavior is usually noisy or data are incomplete (omitted variables or measurement error). McKelvey and Palfrey (1995, 1998) generalizes the notion of Nash equilibrium with noisy best responses, called Quantal Response Equilibrium (QRE). In a QRE, players do not choose the best response with probability one (as in Nash equilibrium). Instead, they choose strategies with higher expected payo¤s with higher probability.

Experimental Economics (ECON3020) Game theory 2 Spring2010 14/28 Quantal Response Equilibrium II

The common parametric model for probabilitic choise is the logit QRE model: for β 0,  exp (βπ ) Pr (Right) = right exp (βπright ) + exp (βπleft ) 1 = , 1 + exp ( β∆) where ∆ represents the payo¤ di¤erence between (Right) and (Left), πright πleft . If β goes to the in…nity, then the QRE converges to a Nash equilibrium. If β goes to zero, the choices become entirely random. (Consistency - equilibrium restriction) Due to that choices are probabilistic, the QRE model assumes that each player has a rational expectation about the value of β, which is in turn consistent with players’choice behavior.

Experimental Economics (ECON3020) Game theory 2 Spring2010 15/28 Treasures and Contradictions

We now look at laboratory data for games that are played only once, which was reported by Goeree and Holt (2001, AER). The treasure is a treatment in which behavior con…rms nicely to predictions of Nash equilibrium or relevant re…nement. (Contradiction) However, a change in the payo¤ structure produces a large inconsistency between theoretical predictions and observed behavior.

Experimental Economics (ECON3020) Game theory 2 Spring2010 16/28 How would you play when R = 5? The unique Nash equilibrium is that each player reports 180.

Traveler’sDilemma Game

Consider the game in which two players simultaneously choose integer numbers between 180 and 300. Both players are paid the lower of the two numbers and, in addition, an amount R > 1 is transferred from the player with the higher number to the player with the lower number. If their numbers are equal, then they just are just paid by that number. For example, if player 1 chooses 210 and player 2 chooses 250, player 1 receives 210 + R and player 2 receives 210 R. How would you play when R = 180?

Experimental Economics (ECON3020) Game theory 2 Spring2010 17/28 The unique Nash equilibrium is that each player reports 180.

Traveler’sDilemma Game

Consider the game in which two players simultaneously choose integer numbers between 180 and 300. Both players are paid the lower of the two numbers and, in addition, an amount R > 1 is transferred from the player with the higher number to the player with the lower number. If their numbers are equal, then they just are just paid by that number. For example, if player 1 chooses 210 and player 2 chooses 250, player 1 receives 210 + R and player 2 receives 210 R. How would you play when R = 180? How would you play when R = 5?

Experimental Economics (ECON3020) Game theory 2 Spring2010 17/28 Traveler’sDilemma Game

Consider the game in which two players simultaneously choose integer numbers between 180 and 300. Both players are paid the lower of the two numbers and, in addition, an amount R > 1 is transferred from the player with the higher number to the player with the lower number. If their numbers are equal, then they just are just paid by that number. For example, if player 1 chooses 210 and player 2 chooses 250, player 1 receives 210 + R and player 2 receives 210 R. How would you play when R = 180? How would you play when R = 5? The unique Nash equilibrium is that each player reports 180.

Experimental Economics (ECON3020) Game theory 2 Spring2010 17/28 Experimental Economics (ECON3020) Game theory 2 Spring2010 18/28 How would you play when c = 0.9?

When c < 1, any common e¤ort in the range (ei = ej ) is a Nash equilibrium. Why?

Minimum-E¤ort

Suppose two players choose “e¤ort” levels, e1 and e2, simultaneously, which can be any integer number between 110 and 170. Each player i’spayo¤ is given by

Ui (ei , ej ) = min ei , ej c ei , for c < 1. f g  How would you play when c = 0.1?

Experimental Economics (ECON3020) Game theory 2 Spring2010 19/28 When c < 1, any common e¤ort in the range (ei = ej ) is a Nash equilibrium. Why?

Minimum-E¤ort Coordination Game

Suppose two players choose “e¤ort” levels, e1 and e2, simultaneously, which can be any integer number between 110 and 170. Each player i’spayo¤ is given by

Ui (ei , ej ) = min ei , ej c ei , for c < 1. f g  How would you play when c = 0.1? How would you play when c = 0.9?

Experimental Economics (ECON3020) Game theory 2 Spring2010 19/28 Minimum-E¤ort Coordination Game

Suppose two players choose “e¤ort” levels, e1 and e2, simultaneously, which can be any integer number between 110 and 170. Each player i’spayo¤ is given by

Ui (ei , ej ) = min ei , ej c ei , for c < 1. f g  How would you play when c = 0.1? How would you play when c = 0.9?

When c < 1, any common e¤ort in the range (ei = ej ) is a Nash equilibrium. Why?

Experimental Economics (ECON3020) Game theory 2 Spring2010 19/28 Experimental Economics (ECON3020) Game theory 2 Spring2010 20/28 Extensive Form Games

As in the ultimatum bargain game, many interesting games are sequential in nature. In sequential games, a …rst mover must try to anticipate how subsequent decision-makers will react. A common approach in game theory is using : begin by considering the …nal decision-maker’schoice, and then work backward to consider the …rst decision-maker’sproblem. The use of backward induction and NE in every eliminates equilibria with threats that are not “credible”. Players must not only have full cognitive capability of conducting multi-stage backward induction but also have a perfect trust on other players’rationality.

Experimental Economics (ECON3020) Game theory 2 Spring2010 21/28 Should You Trust Others to Be Rational?

Experimental Economics (ECON3020) Game theory 2 Spring2010 22/28 Should You Trust Others to Be Rational?

Experimental Economics (ECON3020) Game theory 2 Spring2010 23/28 Should You Believe a Threat that is Not Credible?

The game just considered is a little unusual in that the second player has no reason to punish since the …rst player’s R also bene…ts the …rst player.

Experimental Economics (ECON3020) Game theory 2 Spring2010 24/28 Should You Believe a Threat that is Not Credible?

The game just considered is a little unusual in that the second player has no reason to punish since the …rst player’s R also bene…ts the …rst player.

Experimental Economics (ECON3020) Game theory 2 Spring2010 24/28 Should You Believe a Threat that is Not Credible?

Experimental Economics (ECON3020) Game theory 2 Spring2010 25/28 Classroom Game

Now let’splay a game in a classroom experiment. Go to http://veconlab.econ.virginia.edu/login.htm. Session name is sjc9.

Experimental Economics (ECON3020) Game theory 2 Spring2010 26/28 The

The game you just played is a version of the so-called Centipede Game. The original centipede game (Rosenthal (1982)) has 100 stages that is intended to serve as an extreme stress test of backward induction reasoning process. Centipede games are basically multistage trust games in which, if “Pass” is chosen at each node, the size of pie is growing twice. Thus, the collective gain to passing is huge. The unique subgame perfect NE is choosing “Take” at each node for both players.

Experimental Economics (ECON3020) Game theory 2 Spring2010 27/28 Centipede Game Experiment

The …rst experiment on Centipede game was done by Mckelvey and Palfrey (1992, Econometrica).

Experimental Economics (ECON3020) Game theory 2 Spring2010 28/28