Past exam of the computer science course of the University of Oxford / 2019 / Computational game theory / 1 / e / Solution

Both players can guarantee $k$ by part (d), so at any Nash equilibrium each must receive at least $k$. Their payoffs sum to $2k$, forcing both equilibrium payoffs to equal $k$.

Moreover, an equilibrium action must itself have the maxmin guarantee $k$. If a player's chosen action allowed an opposing response that reduced its payoff below $k$, that response would give the opponent more than $k$, a profitable unilateral deviation. Thus both equilibrium sites belong to the central set identified in part (d).

Conversely, if both choose from that central set, each chosen action guarantees $k$. Neither can improve above $k$, because the other player's fixed central action still guarantees the other player $k$. Consequently, all pure Nash equilibria areFor even $k$, this includes both co-located central profiles and both orders of the two distinct central sites.