The name is absent

In order to save notation let us write x (n) as the strategies of all

-i

players except i in a N.E. with n players, i.e. x (n) = x(n) - x(n). Also

define V ( ) ≡ U (x, x + x ) ≡ V (x, x ). Then, if Proposition 2 a) were
i                 i i -i i i ɪ -i

not true, V (x (п+1), X (n+l))  > V (x (n), x (n))  ≥ V (x (n+l), x (n)).

i i                  -i                        i i              -i                     i i                  -i

Thus, X (n) > X (n+l) which contradicts that x (n) is non-decreasing in n
-i             -i                                                      -i

by Proposition 1 a). In order to show b) let us assume that U (n) = U (n+l).
i                   i

Then, reasoning as above we get x (n) ≥ x (n+l) contradicting that if y > O,
-i            -i

X (n) is strictly increasing in n (by Proposition 1 b)).u
-1

If A. 2 holds but U ( ) is increasing on x we have the reverse conclusion,
i

The following example shows that if A. 2 does not hold, Proposition 2 may fail.

Example 3.- Let us assume 2 agents with identical payoff functions (see Figure
1). Because A.3, payoffs increase in the direction of the arrows. Point A is a
symmetrical N.E. with 2 players since any player can only change unilaterally
x and x on the 45° line (x and x change in the same amount since the
i                                                                        i

strategies of other players are given). By the same token B is a symmetrical

N.E. with 3 players and such that the payoff of 1 and 2 is now greater (notice
that if n = 1 A’A and OA would be identical and the example does not work).

Notice than in Example 3 we have that n > 1. If this is not the case,
i.e. there is a unique incumbent player, the entry of a new player will always

decrease the payoff  of the incumbent, i.  e. her  payoff is bigger under

monopoly than under duopoly as shown by the next Proposition (notice that

Assumptions 1-2 are not required and that if U( ) is increasing in x, it is

easy to show that entry increases the payoff of the incumbent.

18

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