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Minimum Game

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Let’s look at a short series of experiments by one group of experimenters.

• Van Huyck, Battalio and Beil "Asset Markets as an Equilibrium Selection Mechanism: coordination failure, game form auctions, and forward induction." Games and Economic Behavior, 5(3), July 1993, 485-504.

• _____________________"Strategic Uncertainty, Equilibrium Selection, and Coordination Failure in Average Opinion Games,” The Quarterly Journal of Economics, 106(3), August 1991, 885-911.

• _____________________"Tacit Coordination Games, Strategic Uncertainty, and Coordination Failure," American Economic Review, March 1990, 234-248.

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The Minimum Game

Smallest Value 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

Your 5 - - 1.10 0.90 0.70 0.50 0.30 Choice 4 - - - 1.00 0.80 0.60 0.40

of 3 - - - - 0.90 0.70 0.50 X 2 - - - - - 0.80 0.60 1 - - - - - - 0.70

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Procedures• Series of games, after which one will be randomly

chosen to be the payoff period. Today in class, everyone will keep their own accounts. Choices will be collected and reported on the board.

In repeated game experiments, one design decision is whether to pay subjects the sum of their earnings in each period, or choose one period at random to be the payoff period. What are some reasons you might want to do one or the other?

How much information to provide to participants across rounds. The whole distribution? Only the minimum?

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Van Huyck et al. somewhat unusual experimental design.

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After the experiment was completed, a referee noted that the information passed to each player in the original experiment was simply the minimum choice in each period. He asked whether the results would be different if subjects saw the whole distribution (as we did in class…)

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Van Huyck and his colleagues have conducted a number of variations of their

original experiment.

• Median games (payoff is maximized if you choose the median of the numbers chosen)

• Bidding for positions in the minimum game: (Earnings if one of these games is the payoff game will be zero for unsuccessful bidders, and earnings minus bids for successful bidders.

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Median Value of x Chosen

7 6 5 4 3 2 1 7 1.30 1.15 0.90 0.55 0.10 -0.45 -1.10 6 1.25 1.20 1.05 0.80 0.45 0.00 -0.55

Your 5 1.10 1.15 1.10 0.95 0.70 0.35 -0.10 Choice 4 0.85 1.00 1.05 1.00 0.85 0.60 0.25

of 3 0.50 0.75 0.90 0.95 0.90 0.75 0.50 X 2 0.05 0.140 0.65 0.80 0.85 0.60 0.65 1 -0.50 -0.05 0.30 0.55 0.70 0.35 0.70

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Van Huyck, Battalio, and Beil, 1991

* Indicates a mutual best response outcome

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Median game in 1993 auction paper: pi=.10(Median xk) - .05[Median-xi]2+ .60The English Clock auction: 18 subjects bid for 9 rights to participate in

minimum game. Initially, all subjects hold up their bid card. Initial price is 65 cents. Every 5 seconds the price increases by an increment. When the price goes above the price a subject is willing to pay that subject lowers his or her bid card.

Initially the price increment is 5 cents, until 11 subjects remain, then increments are 1 cent. The 9 subjects with raised bid cards pay the price at which the 9th bid card was lowered. These 9 subjects then play the minimum game. (Ties are broken by randomly selecting however many subjects are needed from amongst those subjects lowering their bid card on the final price tick. The price paid equals the price preceding the price tick that resulted in a tie).

The instructions were read aloud to ensure that the description of the game was common information. No preplay negotiation was allowed. After each repetition of the asset market stage game, the marketclearing price, P, was announced. After each repetition of the minimum stage the median action was announced and the subjects calculate their earnings.

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All of a sudden we get 7’s

• What could account for this?

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Could it just be the difference between 9 and 18 players? (9 out of 18: may feel special…)

In experiments 1 to 9 we never observed the payoff-dominant outcome.

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What can we learn from these experiments? What new conjectures do they suggest?

The Minimum game shows

• we don’t always get Pareto optimality; • it takes time to converge to an equilibrium;• size and repetition interact: (2 player

repeated gets to Pareto optimal equilibrium, other conditions do not).

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The Median game shows that – rules matter in a way they wouldn’t if only the equilibrium

structure was important—games with the same equilibria can be very different.

– history can be important. • These two observations go together—if players don’t

reach equilibrium immediately, then games with the same equilibria may be different because they are different out of equilibrium, and players learn from and react to their out of equilibrium experiences. (Another way to think about this is that we may need to consider rational behavior not just in the face of rational behavior by others, but also in the face of boundedly rational or irrational behavior by some members of the population.)

The Auction version of the Median game shows that history can matter in a different way: forward induction, or selection, or perhaps a combination.

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Read

Cachon, Gerard P., and Colin F. Camerer. February 1996. "Loss-Avoidance and Forward Induction in Experimental Coordination Games." Quarterly Journal of Economics, 165-94.