Game Theory and Behavioral Economics
An Illustration: The 21 Coins Game
Rules: In this two person game, the first mover can remove 1, 2, or
3 coins from the initial 21 coins. Then, the other player can remove
1, 2, or 3 coins from those remaining. The players continue to
alternate with the same opportunity (remove 1, 2, or 3 coins). The
winner of the game is the one to remove the last coin.
Questions:
• Do you notice that experience, in the form of playing this game
repeatedly, is a good teacher?
• Do you see how an optimal way to play can be discovered by
“backward induction?”
Theory
• Cooperative Game versus Noncooperative Game: Binding commitments
versus nonbinding commitments
• Extensive Form versus Normal Form of a Game
• Pure strategy versus mixed strategy
• Nash Equilibrium: A set of strategies such that no player has an incentive to
deviate from his or her chosen strategies, given the strategies of others.
• Unique equilibrium versus multiple equilibria
• Roll back equilibrium
Theory
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Focal point: A psychological reason to choose one equilibrium over another
Information Set:
Perfect Information versus Imperfect Information
Subgame Perfect Equilibrium:
Folk Theorem:
State of the World:
Complete Information
Dominated Strategy
Evolutionary Game Theory
Payoff Dominance
Risk dominance
Nash Equilibrium Examples
• Cournot Duopoly: Firm 1 chooses optimal production quantity, given the
quantity chosen by Firm 2
• Bertrand Duopoly: Firm 1 chooses optimal product price, given the price
chosen by Firm 2
• Stackleberg Leader-Follower Duopoly: Leader chooses optimal production
quantity, understanding Firm 2 will choose an optimal production quantity
based upon the quantity chosen by firm 1.
Coordination Game
Player 2
Opera
Player 1
Boxing
Opera
(X,X)
(0,0)
Boxing
(0,0)
(X,X)
Two people prefer to go to the same event together. Because X>0 is assumed, both
players prefer coordinating to not coordinating. A larger value for X indicates more is
at stake.
There are two Nash Equilibria for this game, the two situations where the players
coordinate their actions
Which side of the road should we drive on? In England, they coordinate on the left. In
America, we coordinate on the right.
Battle of the Sexes Game
Player 2
Opera
Player 1
Boxing
Opera
(2X,X)
(0,0)
Boxing
(0,0)
(X,2X)
Two people prefer to go to the same event together. However, in this sexist
interpretation of this game as originally presented, the woman (Player 1) will be better
off if the two coordinate on Opera, while the man (Player 2) will be better off it the two
coordinate on boxing.
An alternative context for this game could be an employee (player 1) employer (player
2) labor market interaction. If employer and employee do not coordinate, so there is no
labor contract, both are worse off. If the two reach a low wage agreement, the
coordination benefits the employer more. If the two reach a high wage agreement, the
coordination benefits the employee more.
Stag Hunt Game
Player 2 (Minimum Effort of Others)
High Effort
Player 1
(Own Effort)
Low Effort
High Effort
(2X,2X)
(0,X)
Low Effort
(X,0)
(X,X)
The Stag Hunt game is a coordination game. However, the two are each better off if
they coordinate on one of the options than the other.
There are still two Nash Equilibria for this game, the two where coordination occurs.
The high effort equilibrium is payoff dominant, meaning it yields a higher payoff to
each player than the low payoff equilibrium.
The low effort equilibrium is risk dominant, meaning each player risks less. In the low
effort equilibrium, a switch in behavior of the other player results in no loss. However,
in the high effort equilibrium, a switch in the behavior of the other player results in a
significant loss.
Prisoner’s Dilemma Game
Player 2
Cooperate
Player 1
Cooperate
Defect
Defect
(4X,4X)
(0,6X)
(6X,0)
(X,X)
Each player (e.g. each prisoner) is better off if they each
cooperate (and do not provide police with information
that incriminates the other), compared to if they each
defect (and provide the information). However,
cooperation is risky, and there is an incentive to defect.
Given each possible strategy of the other, the best
decision is to defect. Thus, the unique Nash Equilibrium
is to defect, implying self interest with complete
knowledge generates inefficiency.
Chicken Game
Player 2
Dare
Player 1
Dare
Chicken
Chicken
(-2X,-2X)
(2X,0)
(0,2X)
(X,X)
Each player (e.g. each nation) hopes to benefit by
challenging the other so that the other “chickens out.”
However, if both choose to challenge the other, then
both are the worst off compared to if they both chicken
out.
Centipede Game
Player 1 and player 2 alternate decisions as long as
the game continues. To continue the game, player 1
must accept risk, but player 1 has an incentive to
continue the game because player 1 benefits from
continuing as long as player 2 chooses to continue
the game. Similarly, player 2 must risk 1 to
continue the game, but player 2 similarly has an
incentive to continue the game because player 2
benefits from continuing as long as player 1 choses
to continue. Using backward induction, one can
deduce that the subgame perfect equilibrium for this
game is for player 1 to end the game on the first
move, which is interesting because player 1 would
be better off by ending it at any later date.
P-Beauty Contest Game
Rules: A group of players are asked to simultaneously pick a number
between zero and 100. The winner is the player who chooses the number
closest to P times the average of the numbers chosen. (Typically P is set to
equal ½ or 2/3)
Why is the game of interest?
• It provides a measure of the player’s depth of thinking
• It demonstrates people will learn when play is repeated
• It demonstrates the idea of deriving a Nash equilibrium by
eliminating dominated strategies.
• It demonstrates the idea that, if other people do not think deeply
enough to play the Nash equilibrium, then it may not best for you
to play the Nash equilibrium
Ultimatum Bargaining Game
Rules: A gain Y is to be split between two players, proposer and
responder. The proposer acts first and proposes how to split the gain,
offering X to the responder. The proposer keeps Y-X. The
responder then decides to either accept or reject the proposal. If the
proposal is accepted the proposal is implemented. If the proposal is
rejected, both proposer and responder receive zero (no deal).
Results: Proposers typically offer more than the minimum, which is
the Nash Equilibrium. Responders will typically reject offers that
approach the minimum, which is also not the Nash Prediction.
Importance: People have objectives not entirely captured by the
direct incentives in the game.
Dictator Game
Rules: A gain Y is to be split between two people, a dictator and a
receiver. The dictator is the only one of the two that makes a choice.
The dictator dictates how the gain is to be split, offering X to the
receiver and keeping Y-X.
Results: Dictators do not typically offer the minimum, which is what
self interest predicts, indicating a preference for “fairness.”
However, dictators do not offer as much as proposers in the
ultimatum bargaining game, an indication that a portion of the offer
in the ultimatum bargaining game is motivated by “gamesmanship,”
rather than being entirely motivated by fairness.
Importance: People seem to have a “social preference,” in addition
to a preference for self.
Investment (Trust) Game
Rules: In a two person game, a first mover decides how much to trust a
second mover by choosing the fraction of an endowment to give to the
second mover. The amount the second mover receives is triple the amount
sent by the first mover, (implying there is a return to investing in trust).
The second mover then has the opportunity to demonstrate trustworthiness
by deciding what fraction of the amount received to send back to the first
mover.
Results: The first movers exhibit trust, even though the Nash equilibrium is
to exhibit no trust. The second movers exhibit trustworthiness, even though
the Nash equilibrium behavior is to exhibit no trustworthiness.
Importance: People seem to be will to accept the risk associated with using
trust to elicit trustworthiness and thereby capture gains that may be
available. This appears to be become people reciprocate trust with
trustworthiness.
Applications
P-Beauty Contest Game
• It captures Keynes view of what is required to predict the stock market
• You do not need to know the market fundamentals, so much as you need to
know what people think is true (on average) about the market fundamentals
• You need to think more deeply than the other thinkers, but not too deeply.
Ultimatum Bargaining and Dictator Games
• While mutually beneficial trade generates a surplus that can be split between
buyer and seller, the trade will be threatened when either party seeks to capture
too large a share of the surplus.
• People will sacrifice self to help enforce “fairness,” roughly defined as being a
“reasonable” split of a gain (where what is reasonable may depend upon
context).
Investment (Trust) Game
• Gift Exchange/Efficiency Wage: Employees and employers that can exhibit
trust and reciprocate with trustworthiness may outperform those who do not.
References
Berg, Joyce, Dickout, John, and McCabe, Kevin (1995). Trust, Reciprocity, and Social
History. Games and Economic Behavior 10, 122-142.
Dixit, Avinash (Summer, 2005). Restoring Fun to Game Theory, Journal of Economic
Education, 205-219.
Forsyth, Robert, Horowitz, Joel, Savin, N.E., and Sefton, Martin (1994). Fairness in
Simple Bargaining Experiments. Games and Economic Behavior 6, 347-369.
Guth, Werner, Schmittberger, Rolf, and Schartz, Bernd (1982). An Experimental Analysis
of Ultimatum Bargaining. Journal of Economic Behavior and Organization 3, 367-388.
Mailath, George (September, 1998). Do People Play Nash Equilibrium? Lessons from
Evolutionary Game Theory. Journal of Economic Literature 36, 1347-1374.
Rabin, Matthew (1993). Incorporating Fairness into Game Theory and Economics,
American Economic Review 83, 1281–1302.
Sexton, Richard (April, 1994). A Survey of Noncooperative Game Theory: Part 1
Theoretical Concepts . Review of Marketing and Agricultural Economics 62(1), 11-28.