- #1
nomadreid
Gold Member
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Suppose we have a game, played in which Alice and Bob play mixed strategies:
(sorry for the dots, but I don't know how to put a table or tab spacing in this text box)
..............Bob
..........Dove, prob q......Hawk, prob (1-q)
...Dove , prob p......(2,3).........(4,5)
Alice
...Hawk, prob (1-p)...(6,7).........(8,9)
Where "prob" indicates the (unknown) probability that Alice or Bob will play that particular move, and a pair (a,b) means the payoff for Alice and Bob, respectively.
To figure out p and q for a Nash equilibrium, which one of the following reasoning procedures is correct:
Alice's payoff, if she plays Dove, is 2q + 4(1-q) = A
Alice's payoff, if she plays Hawk, is 6q + 8(1-q)= B
Bob's payoff, if he plays Dove, is 3p + 5(1-p) = C
Bob's payoff, if he plays Hawk, is 7p + 9(1-p)= D
For a Nash equilibrium, A=B & C=D, so we solve...
...OR
The probabilities of each move are
...............Bob
..........Dove, prob q.....Hawk, prob (1-q)
...Dove , prob p......p*q.......p*(1-q)
Alice
...Hawk, prob (1-p)......(1-p)*q......(1-p)*(1-q)
so that
Alice's payoff, if she plays Dove, is 2pq + 4p(1-q) = A
Alice's payoff, if she plays Hawk, is 6(1-p)q + 8(1-p)(1-q)= B
Bob's payoff, if he plays Dove, is 3pq + 5(1-p)q = C
Bob's payoff, if he plays Hawk, is 7p(1-q) + 9(1-p)(1-q)= D
For a Nash equilibrium, A=B & C=D, so we solve...
Which method (or neither) is correct? Thanks for any indications.
(sorry for the dots, but I don't know how to put a table or tab spacing in this text box)
..............Bob
..........Dove, prob q......Hawk, prob (1-q)
...Dove , prob p......(2,3).........(4,5)
Alice
...Hawk, prob (1-p)...(6,7).........(8,9)
Where "prob" indicates the (unknown) probability that Alice or Bob will play that particular move, and a pair (a,b) means the payoff for Alice and Bob, respectively.
To figure out p and q for a Nash equilibrium, which one of the following reasoning procedures is correct:
Alice's payoff, if she plays Dove, is 2q + 4(1-q) = A
Alice's payoff, if she plays Hawk, is 6q + 8(1-q)= B
Bob's payoff, if he plays Dove, is 3p + 5(1-p) = C
Bob's payoff, if he plays Hawk, is 7p + 9(1-p)= D
For a Nash equilibrium, A=B & C=D, so we solve...
...OR
The probabilities of each move are
...............Bob
..........Dove, prob q.....Hawk, prob (1-q)
...Dove , prob p......p*q.......p*(1-q)
Alice
...Hawk, prob (1-p)......(1-p)*q......(1-p)*(1-q)
so that
Alice's payoff, if she plays Dove, is 2pq + 4p(1-q) = A
Alice's payoff, if she plays Hawk, is 6(1-p)q + 8(1-p)(1-q)= B
Bob's payoff, if he plays Dove, is 3pq + 5(1-p)q = C
Bob's payoff, if he plays Hawk, is 7p(1-q) + 9(1-p)(1-q)= D
For a Nash equilibrium, A=B & C=D, so we solve...
Which method (or neither) is correct? Thanks for any indications.