two grocery stores competing to open new stores at two new locations. store A ha
ID: 3040559 • Letter: T
Question
two grocery stores competing to open new stores at two new locations. store A has resoreces to open 4 stores in three ways that is (3,1),(2,2) and (1,3) .store B has resources to open three stores in 2 ways that is (2,1) and (1,2).assume when A opens x stores B opens y stores in the same location.if x=y each store gets a payoff of zero.if x>y A gets a payooff of y and B gets -y.if x< y ,A gets a payoff of -x and B gets x.
1.represent this as a matrix game
2.find pure strategy nash equilibrium.
3.find the nash equilibrium(ria) in mixed strategies
Explanation / Answer
1.
Below is the payoff matrix for the strategies of A and B.
2.
For A,
when B chooses (2,1) strategy, the best payoff for A is 2 for strategy (3,1.
when B chooses (1,2) strategy, the best payoff for A is 2 for strategy (1,3).
For B,
when A chooses (3,1) strategy, the best payoff for B is 0 for strategy (1,2).
when A chooses (2, 2) strategy, the best payoff for B is -1 for strategy (2,1) or (1,2).
when A chooses (1, 3) strategy, the best payoff for B is 0 for strategy (2,1).
All these strategies are shown in bold.
As, we cam see there are no common strategies of A and B shown in bold. Hence there are no pure strategy nash equilibrium.
3.
As, we can see the strategy 0.5 * (3,1) + 0.5 * (1,3) weakly dominates the strategy (2,2), we can omit the strategy (2,2). The reduced payoff matrix is,
At mixed strategy Nash equilibrium both players should have same expected payoffs from their two strategies.
E[(3,1)] = 2q
E[(1,3)] = 2 (1-q)
E[(3,1)] = E[(1,3)]
=> 2q = 2 (1-q)
=> q = 0.5
E[(2,1)] = -2p
E[(1,2)] = -2 (1-p)
E[(2,1)] = E[(1,2)]
=> -2p = -2 (1-p)
=> p = 0.5
Therefore the mixed strategy Nash equilibrium is:
– Store A: (3,1) with probability 0.5 and (1,3) with probability 0.5,
– Store A: (1,2) with probability 0.5 and (2,1) with probability 0.5,
A B (2,1) (1,2) (3,1) 2+0, -2+0 1-1, -1+1 (2,2) 0+1, 0-1 1+0, -1+0 (1,3) -1+1, 1-1 0+2, 0-2Related Questions
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