Solution
Weak-Link Effort Game
Show the problem again
Two partners each simultaneously choose an integer effort level from 1 to 8. If partner i chooses a_i, then partner i's payoff is 10·min(a_1, a_2) − a_i². How many pure-strategy Nash equilibria (ordered pairs (a_1, a_2)) does this game have?
Worked solution
The answer is 5. First, no equilibrium has a_1 ≠ a_2: the partner choosing the larger number pays more for the same output, so by lowering her effort to match the other she raises her payoff. So equilibria are on the diagonal a_1 = a_2 = k. Raising effort above k only adds cost, because the minimum stays k. Lowering effort to k − 1 changes the payoff from 10k − k² to 10(k − 1) − (k − 1)², a difference of 11 − 2k. Lowering therefore does not pay exactly when 11 − 2k ≥ 0, that is, k ≤ 5. The equilibria are (1,1), (2,2), (3,3), (4,4), (5,5), so there are 5 of them.
Source: Weak-link (minimum-effort) coordination game, as in Van Huyck, Battalio and Beil (1990); payoff function original to AxiomIQ.