Homework # 2 - ISyE 6230 – Economic Decision Analysis Spring ...

Homework # 2 - ISyE 6230 – Economic Decision Analysis
Spring 2009
1. In class we discussed mixed strategies for normal form games and showed a
condition that allows us to solve for the Nash equilibrium. For a two-person, twoaction
normal form game, it is possible to show the equilibria graphically. In this
case, consider the notion of a “response” function. That is, R1 is the best
response that player 1 can make given player 2's action choice. Recall that player
2's action will be the probability of choosing action 1 (and since there are only
two actions, then this defines the probability of choosing action 2), and so the
response function will be a function of the probability choice. For the following
game, make a plot of R1(q) versus R2(p), where p is the probability that player 1
chooses action 1 and q is the probability of player 2 choosing action 1. The
intersection(s) of the the response curves will by definition define the equilibrium
point(s). For the graph, make the x-axis R1(q) (which is the value of p chosen by
player 1) and the y-axis R2(p) (which is the value of q chosen by player 2).
A1
A2
A1 -1,1 1,-1
A2 1,-1 -1.1
2. Repeat this process for the following game:
A1
A2
A1 2,1 0,0
A2 0,0 1,2
3. In class we derived the results for a Cournot duopoly with a linear demand
function. Derive the formula for N firms and compare the results to N firms that
are colluding (think of this as a monopoly that has N even parts). What happens
as N approaches infinity?
4. In class we showed the results for a Bertrand duopoly (but didn’t derive). Explain
why the results hold.
5. Show there are no mixed strategy equilibria in the Prisoner’s dilemma game.
6. (For PhD students). A two-person zero-sum game is one where for each outcome,
the payoffs sum to zero. That is, they are strictly competitive in that for each
outcome there is a winner and a loser. Consider the simple rock, scissors, paper
game. Do the following:
1. Write out the payoff matrix for this game (let 1=win, -1=loss, 0=tie). Note
that since a zero-sum game, you only need a single element for each outcome.
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2. Write a linear program to find the optimal strategy for each player.
3. Comment on the result.
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