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Behavioral eco lect 6: Strategic interaction part 2 (Best responses (Two…
Behavioral eco lect 6: Strategic interaction part 2
Best responses
Both Nash equilibria and subgame-perfect equilibria require players to determine the best respons(es) against the strategies of the others.
Two important components when determining your best response:
(1) your beliefs about what the others will do.
(2) your utility function.
Experimental evidence
In reality players often do not play the Nash or subgame-perfect equilibrium.
Reasons:
(1) Limited strategic reasoning
a player has limited strategic reasoning or
believes that others have limited strategic reasoning.
(2) Utility depends on more than own payoff.
A player’s utility depends on his own payoff and on the payoffs of others.
A player believes that others have utility functions that depend on their own payoffs and the payoffs of others.
Limited strategic reasoning in the Guessing game
Guessing game / Beauty contest game
Please state a number between 0 and 100 (0 and 100 allowed).:
You win if your number is the closest to 2/3 of the mean of all numbers chosen. Your payoff if you win is a fixed predetermined amount. In case of a tie the payoff is equally divided among the winners. The losers receive nothing.
Proof that “all players stating 0” is a Nash equilibrium:
• Assume all other players state 0
• Suppose you state a number x > 0
→ Mean of all numbers = x/n (n = # of players)
→ distance to p times mean:
you: x – p x/n = (n – p) x/n
others: p x/n – 0 = p x/n
→ We have n – p > n – 1 > 1 > p, so the others win.
→ your best response is to state x = 0 so that you win as well.
Let’s consider a more general form of this game:
Players (n>2) state a number between 0 and 100 (0 and 100 allowed). You win if your number is the closest to p times the mean of all numbers chosen. (assuming 0 < p < 1). This game has exactly one Nash equilibrium: All players stating 0.
Proof that here is no other Nash equilibrium:
:
• Suppose you are the one choosing the largest number x > 0.
• Let sum-of-others = S
→ Mean of all numbers = (x + S)/n
→ your distance to p times mean: x – p (x + S)/n = [(n – p)x – pS] /n
→ your best response is to minimize the distance, so to lower x.
→ a person who states the largest number should always lower this number, so stating the largest number x > 0 can never be a best response.
Iterated Dominance in the guessing game...
Guessing game / Beauty contest game
We still consider the more general form of the guessing game:
- Players state a number between 0 and 100 (0 and 100 allowed) -You win if your number is the closest to p times the mean of all numbers chosen. (assuming 0 < p < 1)
Note:
In general the upper bound (100) could be any number larger than 0.