Discontinuous Games and Endogenous Sharing Rules

Beyond chopsticks: Symmetric equilibria in majority auction games

“…6 For an example where the tie-break rule matters, consider the cases where k = n. Here there is a pure-strategy equilibrium for every m at which all bidders bid 1 on all objects if the tie-break rule always breaks n-way ties by naming some one bidder to be the winner of all the objects. For other tie-break rules, when k = n the existence of equilibria can be problematic (but see Section 5 for a reference to the case k = n and m = 1, where it turns out that there is a continuum of symmetric equilibria).…”

“…5 For most of the paper the tie-break rule is immaterial; we will specify one as needed. 6 All bidders desire to maximize their respective expected payoffs.…”

Understanding some recent existence results for discontinuous games

“…2 A unifying approach to both these and the existence results we have focused on can be found in Barelli and Soza (2009). Moreover, our approach also leaves out the existence results of Simon and Zame (1990) and Jackson, Simon, Swinkels, and Zame (2002) for games with endogenous sharing rules. Although it would be interesting to have a sufficiently general existence theorem to include all such results, we stress that our goal is simply to show that several of them can be obtained by an appropriate generalization of classical ideas.…”

“…When discontinuities on players' payoff functions prevent this approach from being used, the classical solution consists in finding conditions that guarantee that the original game can be suitably approximated by a sequence of sufficiently wellbehaved games in the following sense: a fixed point theorem can be applied to the best-reply correspondence of each of the approximating games, and the limit points the resulting Nash equilibria are themselves a Nash equilibrium of the original game (e.g., Dasgupta and Maskin (1986), Simon (1987) and Simon and Zame (1990)). …”

Existence of equilibrium in common agency games with adverse selection

“…However, such discontinuities create no problem for the existence of equilibrium. Indeed, it follows from the above description of a subgame perfect equilibrium that we can regard the family of normal-form games induced by the agent's strategies as a game with an endogenous sharing rule as in Simon and Zame (1990) and, therefore, use their existence theorem to establish the existence of subgame perfect equilibria in menu games. In fact, a vector of menus defines a subset of payoffs for the principals, each of which corresponds to a particular strategy of the agent.…”