Understanding some recent existence results for discontinuous games

Abstract

We introduce a new condition, weak better-reply security, and show that every compact, locally convex, metric, quasiconcave and weakly better-reply secure game has a Nash equilibrium. This result is established using simple generalizations of classical ideas. Furthermore, we show that, when players' action spaces are metric and locally convex, it implies the existence results of Reny (1999) and Carmona (2009) and that it is equivalent to a recent result of Barelli and Soza (2009). Our general existence result also implies a new existence result for weakly upper reciprocally semicontinuous and weakly payoff secure games that satisfy a strong quasiconcavity property.

Approximation results for discontinuous games with an application to equilibrium refinement

“…Note that, by combining Proposition 2 and Corollary 1, we recover Proposition 4 of Carmona [9]. Carmona [10] provides another variation on the theme of better-reply security.…”

“…The aforementioned example in Section 3.3 of Carmona [9] satisfies sequential better-reply security but does not satisfy generalized Bsecurity as a result of Corollary 3.5 in Barelli and Soza [5]. In the next example, we construct a generalized B-secure game that does not satisfy sequential better-reply security.…”

“…In fact, the example of Section 3.3 in Carmona [9] is a compact, quasiconcave game satisfying weak reciprocal upper semicontinuity and weak payoff security (hence sequential better-reply security) for which no Nash equilibria exist.…”

“…It suffices to show that (∆(X i ), u (δ n ,ν) i ) satisfies Condition 6 with respect to the sequence in (9).…”

On the existence of Nash equilibrium in discontinuous games

“…McManus (1964), Roberts and Sonnenschein (1977), Nishimura and Friedman (1981), Topkis (1979), Vives (1990), and Milgrom and Roberts (1990)), some seek to weaken the continuity of payoff functions (cf. Dasgupta and Maskin (1986), Simon (1987), Simon and Zame (1990), Tian (1992aTian ( , 1992bTian ( , 1992cTian ( , 2009), Zhou (1992, 1995), Reny (1999Reny ( , 2009), Bagh and Jofre (2006), Morgan and Scalzo (2007), Carmona (2009Carmona ( , 2011, and…”