CC BY
3Victor V. Rozen
Saratov State University,
Astrakhanskaya St., 83, Saratov, 410012, Russia E-mail: Rozenvv@info.sgu.ru
Abstract A general method for a description of equilibrium points in games with ordered outcomes is proposed. This method is based on a construction of complete family of homomorphisms from a given game with ordered outcomes into games with payoff functions. Using this method, we obtain a description of the set of equilibrium points and Nash equilibrium points for mixed extension of game with vector payoffs. The main result is a finding of equilibrium points in mixed extension of a finite game with ordered outcomes.
Keywords: Equilibrium points, Nash equilibrium, Game with vector payoffs, Game with ordered outcomes, Mixed extension of a game with ordered outcomes.
1. Introduction and Preliminaries
Using a general definition of a game due to Vorob‘ev (1970), we can consider a non-cooperative game in the normal form as a system
G = (N, (Xi)i£N, A, (ui)ieN, F), (1)
where N = {1, ...,n} is a set of players, Xi is a set of strategies of player i, A is a set of outcomes, ui is a binary relation on A, which represents preferences of player i,
F: H Xi ^ A
i£N
is a realization function. The elements of set
X = Xi
i£N
is called situations in game G. For any two situations x,y G X the condition
F(x) ^ F(y) means that outcome F(x) is less preference than outcome F(y) for player i. Subsystem ((Xi)ieN, A, F) forms a realization structure and subsystem (A, (ui)ieN) forms an evaluation structure of game G. In this paper we consider games with ordered outcomes, that is, games in which preference relations are orderings or quasi-orderings.
Recall that an order relation (or partial ordering) on arbitrary set A is called a binary relation ^ on A satisfying the following three conditions for any a, b, c G A:
(a) a ^ a (reflexivity);
(b) a < b, b < c ^ a < c (transitivity);
(c) a < b, b < a ^ a = b (antisymmetry).
A binary relation is called quasi-order if it satisfies the conditions (a) and (b). Note that the linearity condition (that is, a < b or b < a for all a,b G A) in general case need not be satisfied.
Remark 1. It is well known that a problem of construction of a total linear ordering for given local linear orderings have certain difficulties connected with Arroy paradox. These difficulties take place also for the problem of construction of payoff function in the case of many criteria. However, if we consider evaluation structure of some game in the form of orderings or quasi-orderings, these difficulties disappear. For example, using Pareto-dominance we always obtain quasi-ordering on set of outcomes.
It is convenient to study games with ordered outcomes within the framework of the theory of partially ordered sets (for instance, see Birkhoff (1967)).
Let G be a game with ordered (or quasi-ordered) outcomes. Any situation x G X can be given in the form x = (xi)i^N, where xi is the i-th component of x. For xi G Xi, we denote by x || xi a situation whose i-th component is xi and other components are the same as in x. We denote by < the strict part of order <.
Definition 1. A situation x G X is called an equilibrium point in game G, if does not exist such i G N and xii G Xi that the condition
F(x || xi) > F(x) (2)
holds.
Nash equilibrium point is an equilibrium point x for which the outcomes F(x || xi) and F(x) are comparable under order ui for any i G N. In this case it satisfies
Wi
F(x II xi) < F(x). (3)
Remark 2. It is evident that a situation x G X is an equilibrium point if and only if for every i G N, the outcome F(x) is a maximal element in subset {F(x | xi) : xi G Xi} under order ui. If in this assertion "maximal” is replaced by "greatest” then we have a characterization of Nash equilibrium. In the case of linear orderings of outcomes the concepts of equilibrium point and Nash equilibrium point are the same.
The aim of this paper is to give a description of equilibrium points and Nash equilibrium points for games with ordered outcomes of some important classes. Our method is based on conception of complete family of homomorphisms from given game G with ordered outcomes into some class of games with payoff functions. First of all, we introduce the concept of homomorphism for games with ordered outcomes. Let
G = (N, (Xi)ieN, A (ui)ieN, F) and H = (N, (Yi)i^N, B, (<7i)ieN, &)
X =H Xi,Y = n Yi.
ieN ieN
Suppose that for any i G N a mapping i: Xi ^ Yi and a mapping 0: A ^ B are given. Define a mapping ^: X ^ Y by y>(x) = (^i(xi))ieN.
Definition 2. A (n + 1) system of mappings h = (^i,..., n, 0) is called a homomorphism from game G into game H, if the following two conditions are satisfied:
(i) (isotonic condition): for any i G N and ai, a2 G A
(7i
ai ^ a2 =^ 0(ai) ^ 0(a2);
(ii) (commutative condition): 0 o F = & o <p, that is, 0(F(x)) = &(p(x)).
A homomorphism h = (^1,. ..,pn, 0) is said to be strict homomorphism, if 0 is a strict isotonic mapping; homomorphism ”onto”, if each i (i G N) is a mapping ”onto”; an isomorphism, if for any i G N,^i is one-to-one function and mapping 0 is an isomorphism between the ordered sets (A, wi) and (B, ai), that is, the following equivalence
l^i 7 i
ai ^ a2 0(ai) ^ 0(a2) (4)
holds. Obviously, an isomorphism is a strict and ”onto” homomorphism.
Remark 3. Consider two games G and H with ordered (or quasi-ordered) outcomes, in which the sets of strategies for all players are the same. Put
G = {N, (Xi)ieN, A, (^i)ieN, F),H = {N, (Xi)ieN, B, (^i)ieN, &).
Let i be the identity mapping for each i G N, then the mapping ^: X ^ X is the identity mapping also and the commutative condition can be given as follows:
0 o F = &. Thus, in this case, a homomorphism from G into H can be defined as a mapping 0: A ^ B satisfying the following two conditions:
(i) (isotonic condition): 0 is an isotonic mapping of the ordered set (A, wi) into the ordered set (B, ai) for each i G N;
(ii) (commutative condition): 0 o F = &.
Furthermore, if in the condition (i), for any i G N, 0 is a strict isotonic mapping (isomorphism), then 0 is a strict homomorphism (isomorphism) from G into H.
Let K and K be two arbitrary classes of games with ordered outcomes. Fix in these classes certain optimality concepts and let Opt G be the set of optimal solutions of any game G G K, Opt H the set of optimal solutions of any game H gK. If h is a homomorphism from G into H, then a correspondence between outcomes (and also between strategies and between situations) of these games is given; we denote this correspondence also by h.
Definition 3. A homomorphism h is said to be covariant, if h-image of any optimal solution in G is an optimal solution in H, that is, h(Opt G) C Opt H.
A homomorphism h is said to be contravariant, if h-preimage of any optimal solution in H is an optimal solution in G, that is, h-i(Opt H) C OptG.
Now suppose that for each s G S a homomorphism hs of game G G K into some game Hs gK is given.
Definition 4. A family of homomorphisms (hs)sES is said to be covariantly complete if for each x G Opt G there exists such index s G S that hs (x) G Opt Hs.
A family of homomorphisms (hs)sES is said to be contravariantly complete if the condition hs (x) G Opt Hs for all s G S implies x G Opt G.
It is easy to see that the following result is true.
Lemma 1. (a) A family of homomorphisms (hs)sEs is a covariantly complete family of contravariant homomorphisms if and only if
Opt G =U h-i(OptHs). (5)
sES
(P) A family of homomorphisms (hs)sEs is a conravariantly complete family of covariant homomorphisms if and only if
Opt G = Pi h-i(OptHs). (6)
sES
Now consider the case when an optimality concept is the concept of equilibrium. We have
Lemma 2. (a) For equilibrium, strict homomorphisms are contravariant homo-morphisms.
(P) For Nash equilibrium, homomorphisms ”onto” are covariant homomor-phisms.
According to lemmas 1 and 2, for a given game with ordered outcomes, a description of the set its equilibrium points can be reduced to finding of certain covariantly complete family of strict homomorphisms. In fact, such a manner indirectly was used by Shapley (1959) for antagonistic games with vector payoffs. In the present paper we apply this method for description of equilibrium points and Nash equilibrium points of some important classes of games with ordered or quasi-ordered outcomes. In the second section we get some generalizations of results Shapley (1959) for mixed extensions of non-cooperative games with vector payoffs. In the third section we study games with ordered outcomes in the normal form. The first problem for games of this class is an extension of order relation to the set of probability measures. In this paper we use so-called canonical extension introduced by the author
(1976). An effective form of the canonical extension is given in Theorem 4 (see also
corollary 3). The main result of the present work is Theorem 6. This theorem states that there exists an isomorphism from mixed extension of a given game with ordered outcomes into mixed extension of some game with vector payoffs. Combining this result with Theorem 1, we obtain a complete description for the set of equilibrium points of games with ordered outcomes. The last result is illustrated by an example.
2. Games with vector payoffs
An important subclass of games with ordered outcomes forms class of games with vector payoffs. To introduce this class of games it is necessary, as the first step, to fix a certain order relation on IRm. In this paper we consider two following orderings of IRm :
(u1,...,^™) ^ (v1,...,vm) uk ^ vk for all k =1,...,m; (7)
(u1, ...,um) < (vi,...,vm) ^ (ui,...,um) = (vi,...,vm)
or uk < vk for all k = 1,
(8)
Formally, a game G with vector payoffs can be given as follows. Let N = {1,...,n} be a set of players, Xi a set of strategies for player i. Suppose mi be a number of components in vector evaluation of situations by player i. Then the set of outcomes in game G may be regarded as IRmi x ... x IR™ and the realization function F be a mapping from the set of all situations
X = H Xi
iEN
into the set of all outcomes
U IRmi.
iEN
We denote by Fi(x) the i-th component of outcome F (x), i.e.
Fi(x) = (Fi1(x),..., F™(x)) G IRmi. For any i = 1,...,n and x, y G X put
F(x) ^ F(y) ^ Fi(x) ^ Fi(y); (9)
F(x) <i F(y) Fi(x) < Fi(y). (10)
Remark 4. For a game with vector payoffs, we must indicate in the sense of which ordering (9) or (10) preferences of the players are considered. Note also that formally any game with vector payoffs is a game with quasi-ordered outcomes.
m.
For a finite game G with vector payoffs, its mixed extension G is defined as mixed extension of each its component. Namely, in game G a set of strategies for player i is the set X[i, consisting of probability distributions (or probability vectors) Pi over Xi and the realization function F is a mapping from
n X
iE N
into
J]IR ™i
iEN
defined as follows. Given n-tuple of probability distributions
p = (P1,... ,pn) g n Xi;
iEN
as the first step, we define a distribution p(x) over the set of situations
X = Xi
iEN
by setting for any x = (x1,... ,xn) G X : p(x) = p1(x1) • ... • pn(xn). Now we put F{p) = {Fi{p), • • •, Fn{p)), where
^ P(x)Fi1(x),..., ^ p(x)Fm (x)] .
xEX xEX J
Let G be a finite game with vector payoffs. With arbitrary n-tuple of vectors of the type
c =(ci,...,cn) G n !Rmi
iEN
there is associated a numerical game
Gc = N, (Xi)iEN, (Fci )iEN)
where a payoff function for player i = 1,...,n is defined by Fci(x) = (ci, Fi(x)) (we denote by ( , ) the standard scalar product of two vectors). We say that game is the convolution of game G by means of n-tuple of vectors c = (c1,..., cn).
Now we state some results for games with vector payoffs in which preferences for players are given by (17). We first suppose that for game G the set of optimal solutions is the set of equilibrium points in its mixed extension and also for numerical game Gc. We denote by DRm the set of mi-dimensional vectors with strict positive components.
Theorem 1. Let G be a finite game with vector payoffs. Then
(1) For any vector
c =(ci,...,cn) G n IR m
iEN
a mapping
0c: n IRmi ^ IRn
iE N
defined by
0c (u) = 0c(ui, ...,Un) = ((ci, ui), ..., (cn, Un)), (11)
is a contravariant homomorphism of mixed extension of game G into mixed extension of game Gc.
(2) The family of homomorphisms
{0c : c G n IR^}
iE N
is covariantly complete.
F i(p) =
Proof (of theorem 1).
(1) We first prove that 0c is a strict homomorphism. It is enough to verify the strict isotonic condition and the commutative condition (see remark 3). Indeed, assume that for two vectors u = (u1,..., un) and v = (v1,..., vn) G IRmi x ... xIRr u ^i v holds. This means that ui ^ vi, i.e. uk ^ vk for all k = 1,...,mi and uk < vk at least once; because all components of vector ci = (c1,..., cr[li) are positive, we get (ci, ui) < (ci, vi), that is 0c(u) <i 0c(v). It remains to be shown the commutative condition, i.e.
0c o F = FC, (12)
where F is the realization function in mixed extension of game G and Fc is the realization function in mixed extension of game Gc. For any
p g n Xi,
iE N
the *-th component of vector 0C o F(p) is (cj, Fi(p)) and *-th component of vector Fc(p) is
^2p(x)Fci(x).
xEX
Setting ci = (c1, ..., cm), we have for any i = 1,...,n :
(c^FM) = cl J2 p(*)FHx) + ■ ■ ■ + £ P(x)FrH(x) =
xEX xEX
= £ p(x)(ci F/(x) + ... + c™ Fr (x))= E p(x)(ci,Fi(x)) =
xEX xEX
= E p(x)Fci (x)
xEX
which was to be proved. According to Lemma 2(a), we obtain the part (1). The proof of the part (2) is based on the following geometrical fact.
Lemma 3. Suppose P is a polyhedron in IRm which contains 0 and does not contain of semi-positive points (i.e. points u = 0 with u ^ 0). Then there exists, for P at 0, a supporting hyperplane with strict positive normal vector.
It easy to reduce the assertion of lemma 3 to analogous fact for closed cones (see Nikaido (1968), Theorem 3.6). Let us show (7). Let p0 = (p°,...,pn) be an equilibrium point in mixed extension of game G. Denote by Si(p°) a polyhedron generated by the points of the form p0 || xi, xi G Xi : Si(p0) = conv{p° || xi : xi G Xi}. It is easy to show that Si(p°) = {p° \\ Pi '■ Pi & Xi}■ Because Fi is a linear mapping of X1 x ... x Xn into IRmi, it transforms the polyhedron Si(p0) into polyhedron Fi(Si(p0)) C IRm\ From the definition 1 and (9) it follows thatp0 is an equilibrium point if and only if polyhedron Fi(Si(p0)) — F i(jP) does not contain of semi-positive points. Then, by lemma 3, for polyhedron Fi(Si(p0)) — Fi(p°) at 0, there exists a supporting hyperplane with strictly positive normal vector ci G IR^, hence for any i = 1,... ,n we have (q, Fi(p° || pi) — Fi(jP)) ^ 0 and by commutative condition
(12), the last inequality is equivalent to FCi(p° || p{) ^ FCi(p°). Thus, p° is Nash equilibrium point in the mixed extension of game Gc with
c G U IRr
ieN
which completes the proof of Theorem 1. □
By using lemmas 1(a) and 2(a), we have
Corollary 1. Let Eq(G) be the set of equilibrium points in mixed extension of a finite game G with vector payoffs, NEq (Gc) be the set of Nash equilibrium points in mixed extension of game Gc with payoff functions. Then
Eq(G) = \J{NEq(Gc):cel[m,^}. (13)
ieN
Note that in the case of an antagonistic game with vector payoffs it is the result of Shapley (1959).
Now we consider a description of the set of Nash equilibrium points in mixed extension of game G with vector payoffs. As above, we suppose that preferences on vector outcomes are given by (17). We denote by IR^ the set of mi-dimensional vectors with non-negative components.
Theorem 2. Assume G is a finite game with vector payoffs. Then
(1) For any
c G U IRr
ie N
the mapping 0c defined by (11) is a covariant homomorphism from mixed extension of game G into mixed extension of game Gc
(2) The family of homomorphisms
ie N
is contravariantly complete.
Proof (of theorem 2).
A proof of the part (1) is the same as in Theorem 1. To proof the part (2), assume that the situation in mixed strategies
p = (pi,...,pn) G Xi
ieN
is Nash equilibrium point in game Gc for any vector
c G n IRr.
ie N
We have to prove that p is Nash equilibrium in G. Otherwise the condition F(p || qi) Sjj F(p) does not hold for some ieN and </j G X*. By (17) it follows that Fk(p || qi) > Fk(p) for some k = 1,...,mi holds. Take
the vector c = (c1, ...,cn) G IRm1 x ... x IRr where ck = 1 and c{ =0 for
j = 1,..., m-i, j = k. Using commutative condition (12), we obtain
F0i(p II Qi) = (Ci, Fi(p II qi)) = F['(p II qi) > Fi(p) = (a, Fi{p)) = FCiip).
The inequality FCi(p || qi) > FCi(p) is in contradiction with our assumption that p
is Nash equilibrium in game Gc. This completes the proof of Theorem 2. □
By using lemmas 1(p) and 2(^), we get
Corollary 2. Let NEq (G) be the set of Nash equilibrium point in game G, NEq (Gc) the set of Nash equilibrium points in game Gc. Then
NEq(G) = f]{NEq(Gc):ce (14)
ie N
Let us note some results concerning of games with vector payoffs in which preferences for players are given by (8).
Theorem 3. Consider a game with vector payoffs
G = (N, (Xi)ieN, IRmi x ... x IR, (<i)iei, F).
Suppose the set of optimal solutions for G is the set of equilibrium points in its mixed extension, the set of optimal solutions for Gc is the set of Nash equilibrium points in its mixed extension. Then
(1) For any vector
c = (ci, . ..,cn) G IRr
ieN
a mapping 0c defined by (11) is a contravariant homomorphism from mixed extension of game G into mixed extension of game Gc.
(2) The family of homomorphisms
{0c : c G n IRr }
ie N
is covariantly complete.
The proof of the part (1) is like that in Theorem 1. The proof of the part (2) is based on the following known result (for instance, see Nikaido (1968), Theorem 3.5):
Lemma 4. Suppose P is a polyhedron in IRm which contains 0 and does not contain of positive vectors (i.e., vectors u = 0 with u > 0). Then, for P at 0, there exists a supporting hyperplane with non negative normal vector.
3. Games with ordered outcomes
3.1. Canonical extension of order to the set of probability measures
To construct a mixed extension of a game with ordered outcomes, it is necessary the orders, which represent preferences of the players, to extend on set of probability measures. Now we consider such a problem in a general case. In this work, we assume that a set of outcomes A is finite; put A = {a1, ...,ar }, then a probability measure ^
over A can be presented as r-dimensional probability vector j = (j(ai),..., j(ar)), where
j(ak) ^ 0^j(afc) = 1.
k=1
We denote by A the set of all probability vectors over A. Let w be an order relation on A and C0(w) be the set of all isotonic mappings from the ordered set (A, w) into IR. Each f G C0(w) also can be presented as r-dimensional vector: / = (f(a 1); • • • j/(or))- Given a pair (/, /x), where / G Co(uj) and /x G A, define f(ju) as scalar product of these vectors: /(/x) = (/, /x). With any subset H C Cq(uj) we associate an relation wH on set A as follows:
V^(vf gF)7(m)^7H. (is)
The relation wH is said to be an extension of order w on set of probability measures by means of subset H. An extension of w by means of set C0(w) of all isotonic mappings is called a canonical extension of order w and will be denoted by U: uj = iv0'0^). Obviously, Hi is the smallest extension of the form usH with H C C0 (w). We denote by C2 (w) the set of all isotonic mappings from the ordered set (A, w) into two-element set {0, 1}.
Theorem 4. The extensions of order w by means of the set C2 (w) and by means of the set Co(u>) are the same, that is, luc ^ = luc°(w) = ZJ.
The proof of this theorem is based on two following lemmas.
Lemma 5. For any subset H C C0(w) the condition wconeH = wH holds, where
cone H is the convex conic hull of subset H.
Indeed, the inclusion wconeH C wH is evident. Let us prove the converse inclusion. Assume that for two probability vectors /x, v G A the condition /x ^ v holds. It is necessary to show that inequality /(/x) ^ f(v) for any / G coneH holds. Because a mapping f G cone H can be written as
d
f = £ csfs,
s = 1
where fs G H and c,s ^ 0( = 1,...,d), we get
(d \ d
Ylcsfs, jj = £ cs(fs, j) cs(fs
= \J2Csfs’ ^ J = ^ =7(z/)
d
ys=1 / s=1 s=1
ss
ss
Vs=1
which was to be proved.
Lemma 6. Let C+ (w) be the set of all non-negative isotonic mappings of the ordered set (A, w) into IR. Then coneC2(w) = C+(w).
Proof (of lemma 6). Since C+ (w) is an convex cone and C+ (w) D C2(w), we get coneC2(w) C C + (w). To prove the converse inclusion, consider a
function f G C+(w) and let 0 ^ r1 < r2 < ... < rt be all its values. Denote by Ak = {a G A: f (a) ^ rk} and X^ the characteristic function of subset Ak (k = 1,...,t). It is easy to show that the following decomposition of function f holds:
f = r1XA1 + (r2 _ r1)XA2 + ... + (rt _ rt-1)XAt. (16)
Because set Ak satisfies the majorant stability condition, Xak G C2(w) holds, and using (16), we obtain f G coneC2(w). □
Let us prove Theorem 4. The inclusion wC°(w) C wC2(w) is trivial. Now assume that j ^ v under order wC (w). By using lemmas 5 and 6, we ob-
tain that holds /x under order ujc+^\ i.e. /(/x) ^ f(v) for each nonnegative isotonic mapping f from the ordered set (A, w) into IR. Let f0
be arbitrary isotonic mapping from (A, u>) into IR. Then for some positive constant a > 0 we get (/o + a) G C+(u>), hence /o + a(/i) ^ /o + a(z'), i.e. (f0, j) + (a, j) ^ (f0, v) + (a, v). Since (a, j) = (a, v) = a, we obtain fo(/a) ^ foM- This complete a proof of Theorem 4.
u
Corollary 3. The condition j ^ v holds if and only if the inequality j(B) ^ v(B) satisfies for any majorant stable subset B of ordered set (A, w).
(Recall that a subset B C A is called majorant stable in ordered set (A, w) if the
condition: a G B, a' ^ a ^ a' G B holds).
For the proof it suffices to remark that isotonic mappings of an ordered set (A, w) into {0, 1} are exactly the characteristic functions its majorant stable subsets.
3.2. Nash equilibrium points for games with ordered outcomes
Consider a finite game G with ordered outcomes
G = (N, (Xi)ieN, A, (wi)ieN, F). (17)
A mixed extension of game G is defined as a game G of the form
G= {N, (Xi)ie]v, A, (wj)iejv, F} (18)
in which Xj, is the set of probability measures on Xi, A is the set of probability measures on A, ZJi is the canonical extension of order u>i on A and realization function
F is given as follows: for any situation in mixed strategies
P = (Pi,...,Pn) G Xi
ieN
we define a measure F(p) = Fp as the image of the product p1 x ... x pn under F. In a plain form, measure Fp can be given as
Fp(a) = £ p(x) = £ pi(xi) • ...• pn(xn), (19)
F (x)=a F (x)=a
where x = (xi,. ..,xn).
It is evident that Fp(a) is the probability of appearance of the outcome a in situation p. Mixed extension of a game with ordered outcomes also is a game with ordered outcomes.
Let G be a game with ordered outcomes of the form (17); we assume that for any i G N, a mapping 0i: A ^ IR is given. Putting 0 = (01,...,0n), we can construct a numerical game G^ with payoffs functions:
G^ = {N, (Xi)ieN, (0i o F)ieN).
Theorem 5. For any game G with ordered outcomes, the set NEq (G) of Nash equilibrium points in its mixed extension can be presented in the form
NEq (G) = f]{NEq (G^):^ncoN} (20)
ieN
where C0(wi) is the set of all isotonic mappings from the ordered set (A, wi) into IR and NEq (G) is the set of Nash equilibrium points in mixed extension of game G^.
Lemma 7. Denote by 0i o F the payoff function in mixed extension of game G^. Then for any situation in mixed strategies
p = (pi,...,pn) G I} Xi
ieN
the equality
ipi oF(p) = ipiiFp) (21)
holds.
Indeed, setting x = (x1,... ,xn) and p(x) = p1(x1) • ... • pn(xn), we have
0i o F(p) = £ 0i(F(x))p(x) = £ £ 0i(a)p(x) = xeX aeAF (x)=a
= £ V’i(a) £ p(x) = £ tpi(a)Fp(a) = tpi(Fp).
aeA F(x)=a aeA
To prove (21) we note that the condition p0 G NEq (G) holds if and only if for
any i G N measure Fp° is a greatest element in subset {Fp° p : pi G X[i} under order
uJi, that is ip^Fpo) ^ ip^FpO^p.) for any -0® € Co(w*). By lemma 7 this inequality can be represented in the form -0® ° F(p°) ^ ipi ° F (p° || Pi)- Since tfcoF is the payoff function for player i in game G^, the last inequality means that p° is an equilibrium point in Gjf,.
Remark 5. It is evident that for any -0® € Co (w®) function -0® is an isotonic mapping of the ordered set (A, TDi) into IR; since (21), in fact, is the commutative condition, we conclude that mapping 0 defined by ip(n) = (V’i(/■*); • • •> V’n (/■*)) is a homomorphism of game G into game G^ (see Remark 3). From lemmas l(/3) and 7 it follows that the family {0: 0>i G Co(wj), i G N} is a contravariantly complete family of covariant homomorphisms.
3.3. A description of equilibrium points in mixed extension of a game with ordered outcomes
We shall now prove the main result of this paper, Theorem 6. This theorem states an important connection between games with ordered outcomes and simple games with vector payoffs (a game with vector payoffs is called simple, if the components of vector valuation of its outcomes are 0 and 1 only) and also between their mixed extensions. Namely, we prove that mixed extension of a game with ordered outcomes and mixed extension of some simple game with payoff functions are isomorphic; it follows that solutions of these games coincide. Since solutions for a game with vector payoffs in the form of equilibrium points in its mixed extension are given by Theorem 1, we obtain immediately solutions for a game with ordered outcomes.
Consider a finite game with ordered outcomes of the form (17). For each
i = 1,...,n let A1,...,AJm‘i be a list of all majorant stable subsets in the ordered set (A, wi). For any i = 1,...,n define a mapping 9i: A ^ IRmi
by 0i(a) = (0l(a),...,0r[ii (a)), where 9j is the characteristic function of subset
Given a game G with ordered outcomes of the form (17), we construct a simple game Gvect with vector payoffs as follows. In game Gvect strategies for player i is the same as ones in game G and realization function Fvect is defined by
where FVect(x) = (6\(F(x)),..., 9'"li(F(x))). We suppose that preferences for players in game Gvect are presented by (17). As above, we denote by GF mixed extension
Furthermore, we introduce a notion of convolution of a game. Given a game with ordered outcomes G = {N, (Xi)ieN, A, (wi)ieN, F) and an arbitrary n-tuple of functions 0 = (0i, . ..,0n) where 0i: A ^ IR, we can construct a numerical game G^ = {N, (Xi)ieN, (0i o F)ieN), which is called a convolution of game G by means
(see sec. 2); in this case a payoff function of player i is defined as scalar product:
Given a game G with ordered outcomes and a vector ci = (c1,..., cr['"i) g IRmi,
we define a function 0ci: A ^ IR by
Aj C A, j = 1,..., mi, i.e.
0 otherwise.
1 if a G Aj
(22)
F vect(x) = (F^ect(x),..., Fvect(x))
(23)
of game G and by Gvect mixed extension of game Gvect.
of n-tuple function 0. For a game Gvect with vector payoffs, its convolution is defined by means of vector
c =(ci,...,cn) G n JRm<
ieN
Fa (x) = (ci, Fi(x)).
(24)
Aj 3a
where A1,..., Ar[ii is a list of all majorant stable subsets in the ordered set (A, wi).
Theorem 6. Suppose G is a finite game with ordered outcomes of the form (17). Let Gvect be a game with vector payoffs defined above. Then
(1) The convolution of game Gvect by means of vector
c =(ci,...,cn) G n ^
ieN
and the convolution of game G by means of n-tuple of functions 0c = (0ci,..., 0cn) coincide:
Gvcect = G^c. (25)
Moreover for any vector c = (c1, ..., cm) G IR^, function 0ci is strict isotonic under order wi and conversely for any strict isotonic function 0: A ^ IR there exists a vector ci G IRm such that the equality 0 = 0ci holds up to a constant.
(2) Mixed extension of game G and mixed extension of game Gvect are isomorphic: G = (Tect.
Proof (of theorem 6). (1) Indeed, 0ci o F is the payoff function for player i in game G^c. Denote by F£ect the payoff function for player i in game Gvect. For any situation x G X we have
mi
Flect(x) = (ci, Fvect(x)) = £ej(F(x))cj =
j=1
= cl = 0ci (F(x)) = 0ci o F(x).
Aj 3F(x)
Thus, Fvect = 0ci o F which was to be proved. The "moreover part” it follows from lemma:
Lemma 8. Let (A, w) be an arbitrary finite ordered set and A1,..., Am be a list of all its majorant stable subsets. Associate with any vector c G IRm a function 0c : A ^ IR defined by
0c(a) =J2 *.
Aj 3a
Then
(a) For arbitrary c G IRm, function 0c is a strict isotonic mapping of the ordered set (A, w) into IR;
(b) Any strict isotonic mapping 0 of the ordered set (A, w) into IR can be represented up to a constant mapping in the form 0 = 0c for some vector c G IR^-.
Proof (of lemma 8).
(a) Suppose a1 < a2. Then every majorant stable set containing a1 also contains element a2 but converse is false; since all components of vector c are positive, we get 0c(ai) < 0c(a2).
(b) We first consider the case when all values of function 0 are positive. From (16) it follows that for a vector c G IR> whose components are the coefficients of the corresponding XAk, the equality 0 = 0c holds. Let us show that the equality 0 = 0c* holds for some positive vector c* G IRm. Put 00 = 0c°, where c0 = (1,..., 1) G IRm,
then there exists 6 > 0 such that function 0 — 60co remains isotonic and positive. As above, the equality 0 — 60co = 0ci for some vector ci G IR> holds, hence
0 = S0c° + 0ci = 0(sc°+ci). It remains to observe that (Sc0 + c1) G IR> holds. Thus
we obtain the required statement in the case 0 is positive. In the opposite case
consider a mapping 0 + a, where a = maxo£^ |0(a)|, then we have the required
equality up to a constant. □
The proof of the part (2) is based on the following two lemmas.
Lemma 9. For each i = 1,. .., n a mapping Oi'. A —>■ IRm* defined by Oi(n) = (0j (/x), . .., *(/■«)) an isomorphism of the ordered set (A, ZJi) into
the ordered set (IRm% ^).
Indeed, since
ei (M) = £ M(a)ej (a) = £ v-(a) = ^(Aj),
a^A aeAj
by using Corollary 3, we have
Mi M2 (Vj = 1,... ,TOj) /XI (Af) < /x2 (Af)
(yj = 1, • • •, m) dlim) < 1% (/x2) 0*(/xi) ^ 0i(ji2)
which was to be proved.
Lemma 10. Suppose F is the realization function in game G, 'pvect is //je realization function in game ~GveC , then
6o F = Tect. (26)
To prove (26) it suffices to verify it for each component i = 1,...,n. Indeed, for arbitrary situation in mixed strategies
p G n
i£N
we have
03ioF(p)=ei(F(p))=Wi(Fp)=Fp(Ai)= ]T P(x) =
F (x)eAj
= £ P(x) = £ P(x)ej (F(x)).
d{ (F (x)) = 1 x^X
Because the last sum is the j-th component of vector F^ect(p>), we obtain
01 o F(p) = Tiect(p), i.e. 0t o F = hence 0 o F = Tect.
To prove the part (2) of Theorem 6, we observe that (26) is, in fact, the commutative condition for mapping 0; then from Lemma 9 and Remark 3 it follows
that mapping 0 is an isomorphism from game G into game Tfec . This complete
the proof of Theorem 6. □
Corollary 4. Let NEq (G^eC*) be the set of Nash equilibrium points in mixed extension of game Gvcect. Denote by C(wi) the set of strict isotonic mapping of the ordered set (A, wi) into IR. Then according to Theorem 1 and Theorem 6, the set Eq (G) of equilibrium points in mixed extension of game G can be presented as follows:
Eq (G) = U{NEq (Gvrect): C e n ^1; (27)
i£N
Eq (G) = U{NEq (G4): 0 G n C(c*)}. (28)
ieN
Note that equality (28) was first proved by Rozen (1976). The case when G is a finite game with linearly ordered outcomes has been settled by Yanovskaya (1974). It should be noted that statement (28) can be proved also from Theorem A of Aumann (1964).
4. An example
In this section we consider one example which is an illustration of Theorem 6. We construct a game G of two players with ordered outcomes in the following way. Let X = {x1, x2, x3} be a set of strategies for player 1, Y = {y1, y2} be a set of
strategies for player 2 and A = {a, b, c, d} be a set of outcomes. The realization
function F is given by Table 1. The realization structure of game G is represented by the four of objects (X, Y, A, F). Define now evaluation structure of game G as follows. Assume that players evaluate the outcomes under some criteria which they measure in a certain ordinal scale with marks 2 < 3 < 4 < 5. Suppose player 1 use criteria pi, p2, player 2 use criteria q1, q2 and valuations of outcomes by players 1 and 2 are given in Table 2 and Table 3, respectively.
Tablel. Realization function of game G
F yi y2
Xl a d
X2 c b
X3 d c
Table2. Criteria for player 1
Pi P2
a 2 2
b 3 4
c 4 3
d 5 5
Table3. Criteria for player 2
qi q2
a 4 5
b 2 3
c 5 4
d 3 2
In general case, preference relations for players are defined by means of certain decision rules. In this paper, we use Pareto dominance as a basic decision rule. Then preferences of players 1 and 2 are order relations &1, w2 represented by their diagrams in Fig. 1 and Fig. 2, respectively.
d
b
c
a
Figure1. Diagram of order wl
ca
b d
Figure2. Diagram of order
Thus, the evaluation structure of game G is represented by the three of objects (A, ^1, w2). We consider the equilibrium points as optimal solutions of game G.
Obviously, game G have not equilibrium points in pure strategies. Solutions in mixed strategies are founded according to Theorem 6. As the fist step, we need to construct a game with vector payoffs Gvect. Using diagrams of orderings and w2 (see Fig. 1 and Fig. 2), we find all majorant stable subsets in the ordered set (A, w5) (see a) and in the ordered set (A, w2) (see 3):
(a): Ai = {a, b, c, d}, Ai = {b, d}, Ai = {c, d}, Ai = {d}, A5 = {b, c, d}
(3) : a2 = {a b, c}, a2 = {a, C d}, A2 = {a}, A2 = {c}, a2 = {a, c},
A6 = {a, b, c, d}.
Now we define a realization function Fvect of game Gvect by Table 4 according to formula (23).
Table4. Realization function of game Gvect
F vect yi y2
Xl ((1, 0, 0, 0, 0), (1,1,1, 0,1,1)) ((1,1,1,1,1), (0,1, 0, 0, 0,1))
X2 ((1, 0,1, 0,1), (1,1, 0,1,1,1)) ((1,1, 0, 0,1), (1, 0, 0, 0, 0,1))
X3 ((1,1,1,1,1), (0,1,0, 0,0,1)) ((1,0,1,0,1), (1,1,0,1,1,1))
The convolution of game Gvect by means of a vector
c = ((cl, c2, ci, ci, c5), (c1, c2,, c32, c4,, c2, c2)) is a game Gvcect with payoff functions whose realization function is given by the following Table 5.
Table5. Realization function of game Gvcect
F vect yl y2
Xl (cl, ch + c2 + c2 + c2 + c6) (cl + c2 + cl + ci + cl, c2 + c2)
X2 (cl + cf + cl, c1. +c2 + c2 + c2 +c6) (cl +c2 + ci, c2 + c2)
X3 (cj; + cl + cf + cf + Ci, c| + C2) (ci + cf + 4, C2 + C2 + c| + C2 + C2)
Let NEq (GV^C ) be the set of all Nash equilibrium points in the mixed extension of game Gvcect (note that NEq(Gucect) ^ 0 by Theorem of Nash). According to Theorem 6, the set of all equilibrium points in mixed extension of game G is the
--X) ect
set-theoretical union of the family NEq (Gc ), where c = (c5, c2) G IR> x IR>.
References
Vorob‘ev, N. N. (1970). The present state of game theory (in Russian). Uspehi Mat. Nauk, 25, 2 (152), pp. 81-140.
Birkhoff, G. (1967). Lattice theory. Amer. Math. Soc., Coll. Publ., Vol. 25.
Shapley, L. S. (1959). Equilibrium points in games with vector payoffs. Naval Res. Logist. Quart., Vol. 6, 1, pp. 57-61.
Rozen, V. V. (1976). Mixed extensions of games with ordered outcomes (in Russian). Journal Vych. Matem. i Matem. Phis., 6, pp. 1436-1450.
Nikaido, H. (1968). Convex structures and economic theory. Acad. Press, New York and London.
Yanovskaya, E. B. (1974). Equilibrium points in games with non-archimedean utilities (in Russian). Math. Meth. in social sciences, Vilnius, 4, pp. 98-118.
Aumann, B. J (1964). Utility theory without completeness axiom: a correction. Economet-rica, Vol. 32, 1, pp. 210-212.
