Pavlo Prokopovych Presentation SAED 2011

8/6/2019 Pavlo Prokopovych Presentation SAED 2011

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Domain L-Majorization and Equilibrium Existence in Discontinuous Games

by Pavlo Prokopovych (KEI and KSE)

July 1, 2011


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The Single Deviation Property

Consider a compact game G = (Xi; ui)ni=1: Denote X = X1X2: : :Xn

and by EG X the set of pure strategy Nash equilibria of G

Barelli and Soza (2009), Nessah and Tian (2009), Reny (2009):

A game G = (Xi; ui)ni=1 has the single deviation (SD) property if when-

ever x 2 XnEG, there exist a prole of deviation strategies d 2 X anda neighborhood UX(x) ofx such that for all x

0 2 UX(x), there is a playeri for whom ui(di; x

0

i

) > ui(x0)

Both better-reply secure games and diagonal transfer continuous games pos-sess this property


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SDP 1

1:pdf


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Two Remarks about the Single Deviation Property

(1) The assumption that for all x0 2 UX(x), there is a player i for whom

ui(di(x); x0i) > ui(x

0) implies that the set of nonequilibrium points is

open in X

(2) The assumption that a deviation strategy is dened for each player sim-

plies the proof:

Every nonequilibrium point x has an open neighborhood U(x) on whicha well-behaved correspondence Fx = (F1; : : : ; F n) is dened (Fi(z) =di(x) for z 2 UX(x))

On the other hand, having to dene a deviation strategy for each player

makes assumption (ii) less tractable. It is not always necessary


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The Single Deviation Property: Equilibrium Existence

Reny (2009): a three-person quasiconcave game with the SD property having

no Nash equilibrium in pure strategiesIn that example, it is impossible to construct a cover of X for whichassumption (ii) of Corollary 1 (on the convex hulls of deviation strategiesis satised)

Consider a two-player game (X = X1 X2 = [0; 1] [0; 1])

It has the single deviation property, however not every open cover ofX satisesthe assumption on the the convex hulls of deviation strategies


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SDP 3

2:pdf


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The Weak Single Deviation Property

A game G = (Xi; u

i)n

i=1has the weak single deviation (WSD) property if

whenever x 2 XnEG, there exist an open neighborhood UX(x) of x, a

set of players I(x) N = f1; : : : ng, a collection of deviation strategies

fdi 2 Xi : i 2 I(x)g such that, for every x0 2 UX(x)nEG, there exists

i 2 I(x) with ui(di; x0i) > ui(x

0).

The WSD property covers games with a non-closed set of Nash equilibria

(such as second-price sealed-bid auctions)


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The Weak Single Deviation Property: Equilibrium Existence

Theorem 1 Let G = (Xi; ui)i2N be a compact game. Suppose that

(i) G has the weak single deviation property, i.e. for each x 2 XnEG, thereexist an open neighborhoodUX(x) ofx, a set of playersI(x) N, a collection

of pointsfdi(x) 2 Xi : i 2 I(x)g such that, for everyx0 2 UX(x)nEG, there

exists i 2 I(x) with ui(di(x); x0i) > ui(x

0);

(ii) for each A 2 hXnEGi and every z 2 \x2AUX(x), there exists i 2

[x2AI(x) such that zi =2 cof[x2Adi(x)g.Then G has a pure strategy Nash equilibrium.

Theorem 1 follows from a more general result, Theorem 3


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Equilibrium Existence in Two-Player Games

Theorem 2 If a two-player, compact, quasiconcave gameG = (Xi; ui)2i=1

has the weak single deviation property and each Xi is a subset of the realline, then G has a Nash equilibrium.

The proof of Theorem 2 is lengthy since in it a cover satisfying the conditions

of Theorem 1 is constructed

The fact that I(x) may be a proper subset of N plays an important role inthe proof

To check whether assumption (ii) holds, it is enough to check it for every pairof elements of the cover


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SDP 5

3:pdf


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Qualitative Games: Some Denitions

In the description of a game, each payo function can be replaced witha preference correspondence:

Pi : X Xi; Pi(x) = fzi 2 Xi : ui(zi; xi) > ui(x)g:

The corresponding qualitative game is denoted by = (Xi; Pi)i2N

A strategy prole x 2 X is an equilibrium of if Pi(x) = ? for all i

For = (Xi; Pi)i2N; we call the set Dom = [i2NDomPi the domain of (all points that are not equilibria)


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Qualitative Games: Equilibrium Existence

Theorem 3 Let Xi be a nonempty, compact, convex subset of a Hausdor

topological vector space and = (Xi; Pi)i2N be a qualitative game. Sup-

pose, for each x 2 Dom, there exist I(x) N, and an (n + 1)-tuple

(D1x; : : : ; D

nx ; Ux), where D

ix : X

Xi and Ux is an open neighborhood ofx in X, such that

(i) DomDix = Ux and Dix has open lower sections in X for all i 2 I(x), and

DomDix = ? for all i 2 NnfI(x)g;

(ii) for each A 2 hDomi and every z 2 \x2AUx, there exists i 2 [x2AI(x)

such that zi =2 cof[x2ADix(z)g.

Then has an equilibrium.


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Domain L-Majorized Games

If = (Xi; Pi)i2N has no equilibrium and Assumptions (i) and (ii) of

Theorem 3 hold, then it is possible to construct a strict correspondenceF : X X with open lower sections and such that x =2 coF(x) for all

x 2 X (dubbed "of class L" by Yannelis and Prabhakar 1983)

In our terminology, the game with Dom = X is domain L-majorized

It follows from the maximal element existence theorem for correspondences of

class L (Yannelis and Prabhakar 1983, Theorem 5.1) that DomF 6= X,

a contradiction


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Domain L-Majorized Correspondences

There are a number of generalizations of the notion of an L-majorized corre-

pondence (Yannelis and Prabhakar 1983).

A correspondence F : X Y is domain L-majorized if there exists a corre-

spondence F : X Y of class L such that DomF DomF

For = (Xi; Pi)i2N, it is important not to impose the assumption that each

Pi is domain L-majorized (a too strong assumption)

A number of properties of domain L-majorized correspondences are studied


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Two Corollaries of Theorem 3

A generalized version of the Fan-Browder collective xed-point theorem:

Corollary 2 Let X1; : : : ; X n be nonempty, compact, convex subsets of Haus-

dor topological vector spaces, and X = i2NXi. For each i 2 N, letDi : X Xi have open lower sections. If for each x 2 X, there exists

i 2 N such that Di(x) 6= ?, then there exists x 2 X and i 2 N such that

x 2 coDi(x).

Corollary 3 Let each Xi be a nonempty, compact, convex subset of a Haus-dor topological vector space and let = (Xi; Pi)i2N be a qualitative game.

Assume that, for each i 2 N the correspondence Pi : X Xi is domain

L-majorized. Then has an equilibrium in X.


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The Strong Single Deviation Property

A game G = (Xi; ui)i2N has the strong single deviation property if whenever

x 2 XnEG, there exist an open neighborhood UX(x) ofx; a set of indexes

I(x) N; a family of open neighborhoods fUXi(xi) : i 2 I(x)g, a

collection of deviation strategies fdi 2 Xi : i 2 I(x)g; and a number"(x) > 0 such that, for every x0 2 UX(x)nEG, there exists i 2 I(x)

with ui(di; zi) "(x) > ui(x0) for all zi 2 UXi(xi)

Neighborhoods UXi

(xi

), i 2 I(x), and a positive constant "(x) are added

to the denition of the weak SD property


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Better-Reply Security Amended

Lemma 1 A compact gameG = (Xi; ui)ni=1 with no pure strategy Nash

equilibrium has the strong single deviation property if and only if it is

better-reply secure.

Corollary 4 If G = (Xi; ui)ni=1 is compact, quasiconcave, and has the

strong single deviation property, then it possesses a pure strategy Nash

equilibrium.

The better-reply security condition can be amended by considering, insteadof GrG, the nonequilibrium graph of G; GrnG = f(x; u) 2 X Rn j

ui(x) = ui for all i 2 N and x 2 XnEGg


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An Example

Consider the following simultaneous-move timing game between two players

with X1 = X2 = [0; 1] (Reny 1999). The payo to player i is given by

ui(xi; xi) =8>:

1; if xi < xi;

'i(xi); if xi = xi;1; if xi > xi;

where 'i(xi) = 1 if xi = xi and xi < 0:5; and 'i(xi) = 0 if xi = xiand xi 0:5.

The set of Nash equilibria of the game is not compact, therefore, Renys

theorem can not be applied. However the game has the strong single

deviation property


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TG

4:pdf

Pavlo Prokopovych Presentation SAED 2011

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