A selfless person approaches Jones and Smith with a $100 bill and offers to sell it to the highest bidder, but the winning and losing bidders must pay her their bids.
So if Jones bids $2 and Smith bids $1 they pay a total of $3, but Jones gets the money, leaving him with a net gain of $98 and Smith with -$1. If both bid the same amount, the $100 is split evenly between them. Assume that each of them has only two $1bills on hand, leaving three possible bids: $0, $1, or $2. Write out the payoff matrix for this game, and then find its Nash equilibrium.
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The person is willing to give either of Jones or Smith the $100 note based on the amount they bid. Nash equilibrium is the situation where both Jones and Smith are making a decision that gives each of them the best result taking into account the decision of the other.
A payoff matrix for Jones and Smith in the case described in the problem has been drawn below. The amount bid by Jones is in the x-axis and the amount bid by Smith is in the y-axis; the outcome is in the form (gain of Jones, gain of Smith):
0.....(50, 50).....(99, 0)......(98, 0)
1......(0, 99)......(49, 49)....(98, -1)
2......(0, 98)......(-1, 98).....(48, 48)
From the matrix it can be seen that if either of the players bids $2 he does not make a loss irrespective of the amount that the other bids. An optimum situation is when both of them bid $0, in that case both get $50 and the total gain is $100 each but this requires an agreement between the two prior to the bids being placed. The Nash equilibrium is when both the players bid $2. In this case both of them is assured a positive return irrespective of the amount bid by the other though in this case the total amount that they get is $96.
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