Abstract We provide several generalizations of the main results in Reny (1999) for the existence of both pure and mixed strategy Nash equilibria in discontinuous games. We also provide an example demonstrating that one natural additional generalization is not possible.

1. Preliminaries There are N players. Each player i = 1; 2; :::; N has a pure strategy set, Xi ; a nonempty, compact subset of a topological vector space,1 and a payo¤ function ui : X ! R; where X=

N i=1 Xi :

This de…nes a game G = (Xi ; ui )N i=1 :

Throughout, the product of any number of sets is endowed with the product topology. The symbol

i denotes “all players but i.”In particular, X

i

=

j6=i Xj ;

and x

element of X i : The graph of the vector payo¤ function is the subset of X = f(x; u) 2 X

i

denotes an

RN given by

RN j u = (u1 (x); :::; uN (x))g; and cl denotes its closure.

2. Results We present here, without proofs, several results on the existence of Nash equilibria in discontinuous games. Proofs are available from the author upon request and will be included in this paper in a subsequent revision, hopefully by the end of June 2009. The connection to the literature and the relevant references will also be included in a subsequent revision. Financial support from the National Science Foundation (SES-9905599, SES-0214421) is gratefully acknowledged. 1 We follow Royden (1988) and say that a linear vector space V endowed with a topology is a topological vector space if addition is continuous from V V into V; and multiplication by scalars is continuous from R V into V: In particular, unlike some other treatments, this de…nition does not require (although it of course permits) V to be Hausdor¤.

2.1. Pure Strategy Equilibria Throughout this section, we shall assume that for each player i; Xi is convex and that ui (xi ; x i ) is: (a) for each xi 2 Xi bounded below as a function of x

each x

i

2X

i

i

on X i ; 2 and (b) for

quasiconcave in xi on Xi :

A consequence of (a) is that, for each player i; the following is a well-de…ned real-valued function on X : 0

ui (x) = sup U 3x

inf ui (xi ; x i );

i

x0

i 2U

where the sup is taken over all neighborhoods, U; of x i : Moreover, ui (xi ; x i ) is, for each xi 2 Xi ; lower semicontinuous in x

i

on X

i

(see Reny (1999)).

De…nition 2.1. G = (Xi ; ui )N i=1 has the lower single-deviation property if whenever x 2 X is not a Nash equilibrium, there exists x^ 2 X and a neighborhood U of x ; such that for all x0 2 U; there is a player i for whom ui (^ xi ; y i ) > ui (x0 ) for all y 2 U:

Theorem 2.2. If G = (Xi ; ui )N i=1 has the lower single-deviation property, then G possesses a pure strategy Nash equilibrium. Remark 1. Theorem 2.2 generalizes Reny (1999) as the following result states. Theorem 2.3. If G = (Xi ; ui )N i=1 is better-reply secure, then it has the lower single-deviation property. De…nition 2.4. G = (Xi ; ui )N i=1 has the lower …nite-deviation property if whenever x 2 X is not a Nash equilibrium, there exists x1 ; :::; xK 2 X and a neighborhood U of x ;

such that for all x0 2 U; there is a player i and a k such that ui (xki ; y i ) > ui (x0 ) for all y 2 U:

Theorem 2.5. G = (Xi ; ui )N i=1 has the lower …nite-deviation property if and only if it has the lower single-deviation property. De…nition 2.6. G = (Xi ; ui )N i=1 has the conditional lower single-deviation property if whenever (x ; u ) 2 cl and x 2 X is not a Nash equilibrium, there exists x^ 2 X and a neighborhood U

V of (x ; u ); such that for all (x0 ; u(x0 )) 2 U

V; there is a player i for

0

xi ; y i ) > ui (x ) for all y 2 U: whom ui (^

Theorem 2.7. If the players’ payo¤ functions are bounded, then G = (Xi ; ui )N i=1 has the lower single-deviation property if and only if it has the conditional lower single-deviation property. 2

One can in fact allow payo¤s to be unbounded, but this would require introducing the extended reals, since the limiting values of the payo¤ functions will come into play. A more elementary yet equivalent treatment of unbounded payo¤ functions (indeed, even those taking on in…nite values) results by …rst transforming the unbounded payo¤s, vi ; into bounded ones, ui ; via ui = exp vi =1 + exp vi , and then following our treatment.

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2.2. Mixed Strategies We now drop assumptions (a) and (b) and the assumption of convex pure strategy sets from the previous section. Out of the need to calculate expected payo¤s, we shall assume throughout this section that each ui is both bounded and measurable. In addition, we shall assume that each Xi is a subset of a Hausdor¤ topological vector space. Consequently, because the Xi ’s are compact, if Mi denotes the set of (regular, countably additive) probability measures on the Borel subsets of Xi ; Mi is compact in the weak* topology.3 Extend each ui to M = R by de…ning ui (m) = X ui (x)dm for all m 2 M:

N i=1 Mi

Obviously, one can obtain theorems about the mixed extension of G by applying the

results for pure strategy equilibria in the previous section. We instead focus here on an additional result. De…nition 2.8. The mixed extension of G has the …nite deviation property if whenever m 2 M is not a Nash equilibrium, there exist m1 ; :::; mK 2 M and a neighborhood U of

m ; such that for all m0 2 U; there is a player i and a k such that ui (mki ; m0 i ) > ui (m0 ):

Theorem 2.9. If the mixed extension of G has the …nite deviation property, then G possesses a mixed strategy Nash equilibrium. Remark 2. The proof of Theorem 2.9 is straightforward.4 Nonetheless, it does generalize the mixed strategy existence result in Reny (1999) as stated in the next result. Theorem 2.10. If the mixed extension of G is better-reply secure, then the mixed extension of G has the …nite deviation property. (Indeed, one can take K = 1 in De…nition 2.8.)

3. An Example One might hope to improve upon the pure strategy results above in various ways. One possibility would be to replace the lower single-deviation property in Theorem 2.2 with the following single deviation property. 3

This follows from the Riesz representation theorem and Alaoglu’s theorem. See, for example, Dunford and Schwartz (1988). 4 Suppose, by way of contradiction that no Nash equilibrium exists. Then for every m 2 M; each player has …nitely many mixed strategies such that for every m0 in a neighborhood of m; one of these mixed strategies is a pro…table deviation from m0 for some player. The resulting open cover of M has a …nite subcover, by compactness, and so in fact each player has …nitely many mixed strategies –call them deviation strategies – such for every m in M some deviation strategy is a pro…table deviation from m for some player. However, by Nash’s theorem, the …nite game whose pure strategy set is the product of the players’…nite sets of deviation strategies has a Nash equilibrium, producing an element of M that no player can pro…tably deviate from using any of his deviation strategies. This contradiction completes the proof.

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De…nition 3.1. G = (Xi ; ui )N i=1 has the single deviation property if whenever x 2 X

is not a Nash equilibrium, there exists x^ 2 X and a neighborhood U of x ; such that for all x0 2 U; there is a player i for whom ui (^ xi ; x0 i ) > ui (x0 ):

Assume again that each player’s pure strategy set is convex and that (a) and (b) above hold. One might hope that the following is true. FALSE THEOREM. If G = (Xi ; ui )N i=1 has the single-deviation property, then G possesses a pure strategy Nash equilibrium. Unfortunately this theorem is false. The example below has the single deviation property but possesses no pure strategy Nash equilibrium. 3.1. Example. There are three players and each player’s pure strategy set is [0; 1]: For r 2 [0; 1]; de…ne, u0 (r) =

(

0;

if r > 0

1;

if r = 0

u1 (r) =

(

0;

if r < 1

1;

if r = 1

and de…ne

:

The players’payo¤s are de…ned by the following pair of tables.

1:rown2:col x 2 [0; 1=2]

y 2 [0; 1=3]

y 2 (1=3; 2=3)

y 2 [2=3; 1]

(u0 (x); u1 (y); u0 (z)) (u1 (x); u1 (y); u0 (z)) (u1 (x); u1 (y); u1 (z))

x 2 (1=2; 1] (u0 (x); u1 (y); u0 (z)) (u1 (x); u1 (y); u1 (z)) (u1 (x); u1 (y); u1 (z)) 3:table z 2 [0; 1=2]

1:rown2:col x 2 [0; 1=2]

[0; 1=3]

(1=3; 2=3)

[2=3; 1]

(u0 (x); u0 (y); u0 (z)) (u0 (x); u0 (y); u0 (z)) (u1 (x); u0 (y); u1 (z))

x 2 (1=2; 1] (u0 (x); u0 (y); u0 (z)) (u0 (x); u0 (y); u1 (z)) (u1 (x); u0 (y); u1 (z)) 3:table z 2 (1=2; 1]

The entry in each cell determines, for each player, which of the two functions, u0 ( ) or u1 ( ) on [0; 1]; describes his payo¤ as a function of his strategy choice, where the upper table is in e¤ect when player 3’s choice of z is in [0; 1=2] and the lower table is in e¤ect otherwise, and where the …rst coordinate of each entry corresponds to player 1, the second to player 2, and the third to player 3. 4

For example, if x 2 [0; 1=2]; y 2 (1=3; 2=3) and z 2 [0; 1=2]; then this identi…es the cell in

the upper table whose entry is (u1 (x); u1 (y); u0 (z)): Hence, player 1’s payo¤ is zero, because u1 (x) = 0 for all x 2 [0; 1=2]; and player 2’s payo¤ is also zero, because u1 (y) = 0 for all

y 2 (1=3; 2=3). Player 3’s payo¤ may be either zero or one. It is zero if z 2 (0; 1=2] because u0 (z) is zero there, but it is one if z = 0; because u0 (0) = 1:

The players’ strategy sets are compact and convex and their payo¤ functions are quasiconcave in their actions. To see the latter, notice that the functions u0 ( ) and u1 ( ) are quasiconcave on [0; 1] and that, …xing the other player’s strategies a player’s payo¤ as a function of his choice is either determined by u0 ( ) no matter what choice he makes or by u1 ( ) no matter what choice he makes. Next, it is easy to verify that if (x; y; z) is such that y < 2=3; then there is a neighborhood U of (x; y; z) such that for any (x0 ; y 0 ; z 0 ) in U; either player 1 can pro…tably deviate by choosing x^ = 0 or player 2 can pro…tably deviate by choosing y^ = 1 or player 3 can pro…tably deviate by choosing z^ = 0: Thus, (^ x; y^; z^) serves as single deviation in a neighborhood of any such (x; y; z): Similarly, it is easy to verify that if (x; y; z) is such that y > 1=3; then there is a neighborhood U of (x; y; z) such that for any (x0 ; y 0 ; z 0 ) in U; either player 1 can pro…tably deviate by choosing x^ = 1 or player 2 can pro…tably deviate by choosing y^ = 0 or player 3 can pro…tably deviate by choosing z^ = 1: Thus, (^ x; y^; z^) serves as single deviation in a neighborhood of any such (x; y; z): This shows both that the game has the single deviation property and that a Nash equilibrium fails to exist.

Theorem 3.2. Let X be a compact metric space and suppose that g : X ! R is u.s.c., f : X ! R is l.s.c., and g(x)

h : X ! R such that g(x)

f (x) for all x 2 X: Then there exists a continuous funciton

h(x)

f (x) for all x 2 X:

Proof. De…ne the correspondence H : X

R by H(x) = [g(x); f (x)]: Clearly, H is non-

empty-valued, compact-valued and convex-valued. Also, H is l.h.c. because for all a < b; fx 2 X : H(x) \ (a; b) 6= ;g = fx 2 X : g(x) < a and f (x) > bg; is open. Hence, by Michael’s selection theorem (see, e.g., Klein and Thompson, p.98, Theorem 8.1.8), H admits a continuous selection, h : X ! R: QED Corollary 3.3. If X be a compact metric space and f : X ! R is l.s.c., then for every " > 0 there is a continuous function h : X ! R such that for every x 2 X; f (y)

h(x) 5

f (x);

for some y within " of x: Proof. De…ne g(x) = inf f (z); where the in…mum is over all z 2 X whose distance from x is strictly less than ": Then g is u.s.c. and g exists a continuous h such that g(x)

f: Hence, by the previous theorem there

f (x) for all x 2 X: Moreover, for each x 2 X;

h(x)

because f; being l.s.c., achieves a minimum at some point y among all points whose distance from x is less than or equal to "; we have f (y)

g(x)

h(x)

f (x); as desired. QED

References Dunford, N., and J. T. Schwartz (1988): Linear Operators Part I: General Theory. New York: John Wiley and Sons. Reny, P. J. (1999): “On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games,”Econometrica, 67, 1029-1056. Royden, H. L. (1988): Real Analysis. New York: Macmillan.

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