L- INTRODUCTION
In this paper we study the effects of changes in the number of players
and shifts in their payoff function on the strategies played and the payoffs
obtained in a Nash Equilibrium. We will assume on the class of games under
consideration that the payoff function of each player fulfills the following:
I) It can be written as a function of her own strategy (assumed to be one
dimensional) and the sum of the strategies of all players. This assumption has
been called the "Aggregation Axiom" by Dubey, Mas-Colell and Shubik (1980), p.
346 and the corresponding games are called market games (for a different
definition of a market game see Shubik (1984) p. 314). According to M. Shubik
(1984) p. 325) "Games with the above property clearly have much more structure
than a game selected at random. How this structure influences the equilibrium
points has not yet been explored at depth".
II) It satisfies a strong concavity condition slightly stronger than the
Strategic Substitutes case studied by Bulow, Geanakoplos and Klemperer (1985).
The latter implies that the best reply function of each player (i.e. the
mapping selecting the best strategy for a player, given the strategies of the
remaining players) is decreasing on the strategies of other players.
Notice that the class of games satisfying I) and II) is large and include
a) Models of strategic competition in quantities (as the Cournot model,
competition under rationing schemes -see Romano (1988)-, etc), i.e. oligopoly
without or (in some cases) with product differentiation,