1 Economics 776 Experimental Economics First Semester 2007 Topic 5: Coordination Games Assoc. Prof. Ananish Chaudhuri Department of Economics University

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Economics 776Experimental Economics

First Semester 2007Topic 5: Coordination Games

Assoc. Prof. Ananish ChaudhuriDepartment of Economics

University of Auckland

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Battle of the Sexes

Opera Boxing

Opera 2, 1 0, 0

Boxing 0, 0 1, 2

Three equilibria

A third equilibrium in mixed strategies

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Stag-Hunt Games (Rousseau – A Discourse on Inequality, 1755)

Stag Rabbit

Stag 8, 8 0, 5

Rabbit 5, 0 5, 5

Three equilibria

PayoffDominant

Secure/Risk DominantA third equilibrium in mixed strategies

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Examples of Coordination Problems

• Work norms• Location of industries – economic geography• Development of urban neighbourhoods• Adoption of technology/diffusion of innovations

– QWERTY keyboard• Contributions to public goods

– high contributions “efficient” equilibria while low contributions “inefficient equilibria” (Rabin, 1993)

• Bryant’s (1983) Keynesian model with a continuum of underemployment equilibria

• Diamond and Dybvig (1983) model of bank runs

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Selection Criteria in Coordination Games - Cooper, Dejong, Forsythe and Ross (1990)

• Is the Nash equilibrium selected?

• Is the Payoff-dominant Nash Equilibirum selected?

• Do dominated strategies make a difference?

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Selection Criteria in Coordination Games - Cooper, Dejong, Forsythe and Ross (1990)

Hypotheses:

• The Pareto-dominant Nash equilibrium will be selected

• Dominated strategies are irrelevant to equilibrium selection

• The outcome will be a Nash equilibrium

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1 2 3

1 320, 320 440, 420 500, 180

2 420, 440 600, 600 660, 360

3 180, 500 360, 660 420, 420

Column Player

RowPlayer

Game 1

Symmetric game; strategy 2 is a dominant strategy for both players; Nash equilibrium (2, 2)

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1 2 3

1 525, 20 555, 60 585, 0

2 505, 110 625, 420 700, 495

3 385, 200 550, 645 625, 720

Column Player

RowPlayer

Game 2

Unique Nash equilibrium at (2, 3)

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1 2 3

1 350, 350 350, 250 1000, 0

2 250,350 550, 550 0, 0

3 0, 1000 0, 0 600, 600

Column Player

RowPlayer

Game 3

Two Nash Equilibria at (1, 1) and (2, 2); latter is Payoff dominant; Joint maximum at (3, 3) but strategy 3 is dominated by strategy 1

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1 2 3

1 350, 350 350, 250 700, 0

2 250,350 550, 550 0, 0

3 0, 700 0, 0 600, 600

Column Player

RowPlayer

Game 4

Two Nash Equilibria at (1, 1) and (2, 2); latter is Payoff dominant; Joint maximum at (3, 3) but strategy 3 is dominated by strategy 1

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1 2 3

1 350, 350 350, 250 700, 0

2 250,350 550, 550 1000, 0

3 0, 700 0, 1000 600, 600

Column Player

RowPlayer

Game 5

Two Nash Equilibria at (1, 1) and (2, 2); latter is Payoff dominant; Joint maximum at (3, 3) but strategy 3 is dominated by both 1 and 2

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1 2 3

1 350, 350 350, 250 700, 0

2 250,350 550, 550 650, 0

3 0, 700 0, 650 600, 600

Column Player

RowPlayer

Game 6

Two Nash Equilibria at (1, 1) and (2, 2); latter is Payoff dominant; Joint maximum at (3, 3) but strategy 3 is dominated by both 1 and 2

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• In Games 5 and 6 the threat of getting zero from failed coordination has been removed– In Games 3 and 4 if one player chooses 2 while the

other chooses 3 then both players get zero

• In Games 5 and 6 the risk of a bad payoff from choosing strategy 2 (the strategy that is commensurate with the payoff dominant outcome) has been eliminated

• This might make the play of strategy 2 and the (2, 2) outcome more likely in Games 5 and 6.

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Interpretation

• These results provide evidence against Pareto-dominance as a selection criteria for coordination games

• Indicate that cooperative, dominated strategies influence equilibrium selection

• At some point during the play of these games, some participants placed positive probability on an opponent playing the cooperative, dominated strategy

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Interpretation

• Is strategy 3 important because it is the cooperative strategy or do players place prior probability on a dominated strategy even when it does not support the cooperative joint maximum?

• That is a player might believe that his opponent will be “cooperative” or will be “irrational” where the latter term refers to the play of a dominated strategy by a self-interested player

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1 2 3

1 350, 350 350, 250 700, 0

2 250,350 550, 550 0, 0

3 0, 700 0, 0 500, 500

Column Player

RowPlayer

Game 7

Two Nash Equilibria at (1, 1) and (2, 2); latter is Payoff dominant; similar to Game 4

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1 2 3

1 350, 350 350, 250 1000, 0

2 250,350 550, 550 0, 0

3 0, 1000 0, 0 500, 500

Column Player

RowPlayer

Game 8

Two Nash Equilibria at (1, 1) and (2, 2); latter is Payoff dominant; similar to Game 3

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Interpretation

• The cooperative outcome now is (2, 2)• If players place the same prior probability on

their opponent playing the dominated strategy in Game 7 (8) as they did in Game 4 (3) the outcome observed in these two games should be the same

• If players place positive probability on their opponent playing the cooperative strategy rather than the dominated strategy players should expect that their opponent is more likely to play strategy 2 in Game 7 (8) as opposed to Game 4 (3).

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• Thus, when strategy 3 no longer supports the cooperative outcome players evidently place a lower probability on its paly by their rival.

• In Games 3 - 6 the cooperative dominated strategy is chosen 11% of the time while in Games 7 – 8 it is played only 1.8% of the time.

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The Minimum Effort Game (Van Huyck, Battalio and Beil, AER, 1990)

Smallest Value of X Chosen

YourChoice

Of X

7 6 5 4 3 2 1

7 1.30 1.10 0.90 0.70 0.50 0.30 0.10

6 -- 1.20 1.00 0.80 0.60 0.40 0.20

5 -- -- 1.10 0.90 0.70 0.50 0.30

4 -- -- -- 1.00 0.80 0.60 0.40

3 -- -- -- -- 0.90 0.70 0.50

2 -- -- -- -- -- 0.80 0.60

1 -- -- -- -- -- -- 0.70

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The Minimum Effort Game

(Van Huyck, Battalio and Beil, AER, 1990) • Coordination Game where each subject in a

group of N subjects chooses an integer, ci from the set {1, 2, 3, 4, 5, 6, 7}.

• Individual payoffs are determined for each subject by the payoff function

i = k + a[ min {c1…cn} ] – bci

• Choosing k = $0.60, a = $0.20 and b = $0.10 defines the game depicted above

• Sometimes called a “weak-link” game

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The Minimum Effort Game (Van Huyck, Battalio and Beil, AER, 1990)

• The Nash equilibria are located along the diagonal

• They are Pareto ranked with the highest payoff ($1.30) occurring when all players choose 7

• But choosing 1 is the “secure” action since a player choosing 1 is guaranteed $0.70

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Interpretation

• In large groups it seems subjects assign positive probability (albeit small) that someone will choose zero and this small probability results in coordination failure

• Because even if you think one person will choose “1” then your best response is clear – to choose “1” as well

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Conclusions

• The inefficiency is primarily due to “strategic uncertainty”– Some subjects conclude that it is too risky to

choose the strategy that leads to the payoff dominant outcome

• Deductive selection principles do not necessarily enable participants to coordinate to the payoff dominant equilibrium

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Conclusions

• While participants seem able to coordinate to a Nash equilibrium, it is not given that they will be able to coordinate to the Payoff dominant outcome

• This then leads us to look for mechanisms that may facilitate such coordination

• This is what we turn to next

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Communication in Coordination Games - Cooper, Dejong, Forsythe and Ross (1992)

• Explore the role of communication - “cheap talk” - in facilitating coordination

• Two games– Cooperative coordination game– Simple Coordination game

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Cooper, Dejong, Forsythe and Ross (1992)– Cooperative Coordination Game (CCG)

1 2 3

1 350, 350 350, 250 1000, 0

2 250,350 550, 550 0, 0

3 0, 1000 0, 0 600, 600

Row player

Column player

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Cooper, Dejong, Forsythe and Ross (1992) – Simple Coordination Game (SCG)

1 2

1 800, 800 800, 0

2 0, 800 1000, 1000

Row player

Column player

Risk dominant

Payoff Dominant

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• Coordination failure results from the strategic uncertainty about the choices of others

• Thus allowing players to communicate might facilitate coordination by fostering optimistic beliefs about the play of a particular strategy.

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• If communication works to select an equilibrium then communication could resolve any coordination failures observed in games without cheap talk

• The coordination game with communication is a two-stage game between two players

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• In the first stage, player(s) communicate by sending messages to one another

• In the second stage actions are chosen

• Since the payoffs are independent of the actual messages, the messages are essentially “cheap talk”

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• Two alternative communication structures are examined

– One way communication where only one of the two players can send a message

– Two-way communication with both players being able to send messages, simultaneously

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• Subjects play a sequence of one-shot games against anonymous opponents

• Actual game preceded by messages

• We look at the SCG first, followed by the CCG

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• Three conclusions:– In the game without communication, play of

the (1, 1) equilibrium is observed– One way communication increases the

frequency of the (2, 2) equilibrium, but a significant number of coordination failures were observed

– Two-way communication resolves coordination failures; strategy 2 is played almost all the time

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• In the CCG, equilibrium (2, 2) is not always observed with one-way or two-way communication

• In the one-way communication treatments, strategy 2 is not always announced

• Announcements of 1 and 2 are generally followed by play of (1, 1) and (2, 2) respectively

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• Over half the announcements in two-way communication are different from strategy 2

• Announcements of (2, 2) are generally followed by play of (2, 2)

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Credible Assignments In Coordination Games – Van Huyck, Gillette and Battalio

(1992)• A strict equilibrium in a game is defined

as an assignment to each player of a strategy that is a unique best response for him when the others use the strategies assigned to them

• Equilibrium analysis often fails to determine a unique outcome in many games and fails to determine rational behaviour or predict the outcome

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(1992)• One conjecture is that players use

deductive principles to solve the coordination problem

• A popular selction principle involves an appeal to pre-play communication

• An “arbitrator”/ “mediator”/ “referee” exists with the ability to make common information assignments to all the players in the game

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(1992)• Subjects divided into row and column

players• Each row and column pair meet only once• Anonymous pairings• An external arbiter makes “common

knowledge” assignments– Announcements read out loud and also

projected on the front wall of lab where all subjects could see the announcement

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(1992)• ROW CHOOSE ROW AND COLUMN CHOOSE

COLUMN . IF THE ROW PARTICIPANT CHOOSES ROW THEN THE BEST THE COLUMN PARTICIPANT CAN DO IS TO CHOOSE COLUMN . IF THE COLUMN PARTICIPANT CHOOSES COLUMN THEN THE BEST THAT THE ROW PARTICIPANT CAN DO IS TO CHOOSE ROW .

• NOTICE, from the payoff matrix, that if both the Row and Column participants follow the message then they both earn 8 experimental dollars. However, if one of the participants follows the message and the other does not, then both participants will earn a smaller amount.

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(1992)• Even without an arbiter a majority of the

pairs implement the payoff dominant equilibrium

• Assignment to (1, 1) is implemented by 98% of the pairs

• Assignments to (2, 2) and (3, 3) are less credible and implemented by only 25% and 17% of the pairs

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(1992)• However one crucial difference between

this paper and Cooper et al.’s (1990) finding is that is that in this paper the payoff-dominant outcome is also (probably?) risk dominant

• Thus the implementation of the (1, 1) equilibrium may not be terribly surprising

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(1992)• Without an arbiter 70% of the pairs

implemented the equal-payoff equilibrium in the asymmetric coordination game

• An assignment to (3, 3) – the preferred outcome for the column player – was credible to only 16% of the pairs

• Of the 58 row defections, 57 were to (2, 2) and of the 49 column defections, all 49 were to (2, 2) the equal-payoff outcome

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Bangun, Chaudhuri, Prak and Zhou (2006)

• In the Van Huyck, Gillette, Battalio (1992) game there is no obvious conflict between payoff dominance and risk dominance

• The (1, 1) outcome is the most credible outcome in that game

• Will an assignment still work when there is a conflict between payoff and risk dominance?

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Payoff dominant Risk dominant

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• Three treatments:

1. No assignment2. Almost common knowledge of

assignment1. Announcement made public but not read

aloud

3. Common Knowledge of assignment1. Announcement made public and read aloud

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Loss Avoidance and Forward Induction in Coordination Games – Cachon and

Camerer (1996)• Introduces new selection principle “loss

avoidance”

• Players do not pick strategies that result in certain losses for them

• People only pick (and expect others to pick) strategies that will result in gains

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Camerer (1996)• Two important features of this game• First, players’ payoff are decreasing with

the (absolute) difference between their own action and the median. Hence a player’s best action is to choose the number the player believes will be the median

• Second, the seven pure strategy Nash equilibria are Pareto ranked

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Camerer (1996)• Now suppose the players have to pay a

fee of 225 to play the game

• This is optional

• Subjects who opt out earn nothing

• In this case loss avoidance is coupled with a stronger selection principle – forward induction

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Camerer (1996)• In the presence of the “fee”, a subject’s decision

to play the game, rather than opt out, implicitly communicates the subject’s expectations concerning the outcome of the game

• A subject will not choose an action than guarantees a lower payoff than what could be earned for sure by opting out

• Forward induction improves coordination because it shrinks the set of plausible equilibria in a game

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Camerer (1996)• Loss avoidance and forward induction are

closely linked• Loss avoidance applies if players assume

that others will avoid certain losses• Forward induction applies if players

assume that others will avoid an opportunity loss by choosing an equilibrium which is better for them than an option they chose to forgo

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Camerer (1996)• In many games the two principles lead to

the same set of selected strategies• To isolate the effectiveness of loss

avoidance C&C study two kinds of median action games

• In games with the option to opt out coordination improved which meant that the combination of forward induction and loss avoidance selected better equilibria

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Camerer (1996)• But improved coordination is also

observed in games with no options and possible losses, where forward induction did not apply and loss avoidance did

• C&C conclude that loss avoidance could explain some of the improvement in coordination which previously had been attributed exclusively to forward induction

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Camerer (1996)• 9 subjects• Publicly known cost to play the game• Players can opt out

– But this implies that number of players decrease which may itself facilitate coordination

• As the entry fee is increased, forward induction eliminates more and more of the 7 Nash equilibria facilitating coordination on the highest action equilibria

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Camerer (1996)• 4 sessions of “must play”

• 4 sessions of “opt out”

• 9 rounds

• First three rounds are indentical with zero entry cost and no “opt out”

• Entry fee raised to $1.85 in rounds 4 – 6

• Entry fee further raised to $2.25 in rounds 7 - 9

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Camerer (1996)• Forward induction then suggests that subjects

who opt to play in the “opt out” condition will then choose 6 or 7 (which pays $2.40 or $2.60 in equilibrium) when the entry fee is raised to $2.25

• Forward induction predicts nothing in the “must play” condition

• Loss avoidance predicts that subjects will play 6 or 7 in both conditions since playing 5 yields only $2.20 in equilibrium not enough to cover the entry cost of $2.25

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Camerer (1996)• The crucial difference comes between

round 6 and 7 in each session

• The null hypothesis that the distribution between actions differences between round 6 and 7 is the same can be rejected

• The differences however are not very strong

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Camerer (1996)

• Conclusion: Loss Avoidance, absent forward induction, does not generate precisely the same changes in median actions (and improvement in coordination) that forward induction does but the changes are similar.

Documents

1 Economics 776 Experimental Economics First Semester 2007 Topic 5: Coordination Games Assoc. Prof. Ananish Chaudhuri Department of Economics University