CC BY
2Cooperative Optimality Concepts for Games with Preference Relations
Tatiana F. Savina
Saratov State University,
Faculty of Mechanics and Mathematics,
Astrakhanskaya St. 83, Saratov, 410012, Russia E-mail: suri-cat@yandex.ru
Abstract. In this paper we consider games with preference relations. The cooperative aspect of a game is connected with its coalitions. The main optimality concepts for such games are concepts of equilibrium and acceptance.
We introduce a notion of coalition homomorphism for cooperative games with preference relations and study a problem concerning connections between equilibrium points (acceptable outcomes) of games which are in a homomorphic relation. The main results of our work are connected with finding of covariant and contrvariant homomorphisms.
Keywords: Nash equilibrium, Equilibrium, Acceptable outcome, Coalition homomorphism
1. Introduction
We consider a n-person game with preference relations in the form
G = {(Xi)ieN ,A,F, (pi)i€N) (1)
where N = {1, . ..,n} is a set of players, Xi is a set of strategies of player i (i G N), A is a set of outcomes, realization function F is a mapping of set of situations X = Xi x ... x Xn in the set of outcomes A and pi C A2 is a preference relation of player i. In general case each pi is an arbitrary reflexive binary relation on A.
Pi
Assertion a1 < a2 means that outcome a1 is less preference than a2 for player i. Given a preference relation pi C A2, we denote by pf = pi n p-1 its symmetric part and p* = pi\pf its strict part (see Savina, 2010).
The cooperative aspect of a game is connected with its coalitions. In our case we can define for any coalition T C N its set of strategies XT in the form
Xt =n Xi. (2)
ieT
We construct a preference relation of coalition T with help of preference relations of players which form the coalition. We denote a preference relation for coalition T by pT. The following condition is minimum requirement for preference of coalition T:
PT Pi
ai ^ a2 ^ (Vi G T) ai ^ a2. (3)
In section 2 we consider some important concordance rules. Let K be a fix collection of coalitions. In section 3 we introduce the following cooperative optimality concepts: Nash K-equilibrium, K-equilibrium, quite K-acceptance, K-acceptance and connections between these concepts are established in Theorem 1. In next section we consider coalition homomorphisms. The main results of our paper are presented in section 5.
2. Concordance rules for preferences of players
To construct a preference relation for coalition T we need to have preference relations of all players its coalition and also certain rule for concordance of preferences of players. Such set of rules is called concordance rule. It is known that important concordance rules are the following.
2.1. Pareto concordance
Definition 1. Outcome a2 is said to (non strict) dominate by Pareto outcome a\ for coalition T if a2 is better (not worse) than ai for each i G T, i.e.
In this case symmetric part of preference relation for coalition T is defined by the formula
Thus, outcome a2 dominate ai if and only if a2 is better than ai for all players of coalition T and strictly better at least for one player j G T.
2.2. Modified Pareto concordance
In this case strict part of preference relation pT is defined by the equivalence
2.3. Concordance by majority rule
Outcome a2 is strictly better than outcome a\ for coalition T if and only if a2 is strictly better than ai for majority of players of coalition T, i.e.
For this rule, symmetric part of preference relation pT is given by the equivalence
PT
Pi
(4)
ai P'T a2 ^ (Vi G T) ai pPii a2
(5)
and strict part is defined by the formula
Pi
PT
ai < a2 ^
(Vi G T) ai < a2, (3j G T) ai < a2
(6)
Pt Pi
ai < a2 ^ (Vi G T) ai < a2,
(7)
and symmetric part is given by
ai P^T a2 ^ (Vi G T) ai ~ a2.
(8)
Pt
a2 > ai ^
2.4. Concordance under summation of payoffs
For games with payoff functions in the form H = ((Xi)ieN , (wj)ieN), the following concordance rule of preferences for coalition T is used
Pt
x1 < x2 ui(xi) < Ui(x2) (9)
ieT ieT
and the strict part of pT is given by:
x1 < x2 ui(xi) <^^ui(x2).
ieT ieT
In this case preference relation pT and its strict part are transitive.
Remark 1. Let {Ti,..., Tm} be partition of set N. Then collection of strategies of
these coalitions (xTl,... ,xTm) define a single situation x G X in game G. Namely, the situation x is such a situation that its projection on Tk is xTk (k = l,...,m).
Hence we can define a realization function F by the rule: F (xTl,..., xTm) = F (x). In particular if T is one fix coalition then the function F (xT, xN\T) is defined.
Remark 2. Consider a game with payoff functions H = ((Xi)^N , (ui,)ieN) where ui: ^n Xi ^ IR is a payoff function for players i. Then we can define the prefer-
ence relation of player i by the formula
Pi
x1 ^ x2 ^ ui(xi) < ui(x2).
Let the preference relation of coalition T be Pareto dominance, i.e.
Pt
x1 < x2 (Vi G T) ui(xi) < ui(x2).
Then considered above concordance rules is becoming well known rules for cooperative games with payoff functions. (see Moulin, 1981).
3. Coalitions optimality concepts
In this part we consider games with preference relations of the form (1). For games of this class two types of optimality concepts are introduced and connections between these concepts are established.
Let K be an arbitrary fixed family of coalitions of players N.
3.1. Equilibrium concepts
Definition 2. A situation x0 = (x0)ieN G X is called Nash K-equilibrium (Nash K-equilibrium point) if for any coalition T G K and any strategy xT G XT the condition
Pt
F(x0 || xt) < F(x0) (10)
holds.
Remark 3. 1. In the case K = {{1} ,..., {«■}}, Nash K-equilibrium is Nash equi-
librium in the usual sense.
2. In the case K = {N}, a situation x0 is Nash {N}-equilibrium means F (x0) is greatest element under preference pT.
We now define some generalization of Nash equilibrium.
A strategy x<T G Xt is called a refutation of the situation x G X by coalition T if the condition
F(x || xT) > F(x) (11)
holds.
Definition 3. A situation x0 = (x0)ieN G X is called K-equilibrium point if any coalition T G K does not have a refutation of this situation, i.e. for any coalition T gK and any strategy xT G XT the condition
T
F(x0 || xt) T F(x0)
holds.
Remark 4. 1. In the case K = {{1},..., {«■}}, K-equilibrium is equilibrium in
the usual sense.
2. In the case K = {N}, K-equilibrium point is Pareto optimal.
3. In the case K = 2N, K-equilibrium point is called strong equilibrium one.
3.2. Acceptable outcomes and acceptable situations
A strategy xT G XT is called a objection of coalition T against outcome a G A if for any strategy of complementary coalition xN\t G XN\t the condition
T
F(xt,xn\t) > a (12)
holds.
Definition 4. An outcome a G A is called acceptable for coalition T if this coalition does not have objections against this outcome.
An outcome a G A is said to be K-acceptable if it is acceptable for all coalitions T gK, that is
T
(VT G K)(Vxt G Xt)(3xn\t G Xn\t)F(xt,xn\t) T a. (13)
A strategy x<T G Xt is called a objection of coalition T against situation x* G X if this strategy is an objection against outcome F (x*) .
We define also a quite acceptable concept by changing quantifiers: VxT 3xN\t ^ 3xn\t Vxt .
Definition 5. An outcome a is called quite K-acceptable for family of coalitions K if the condition
T
(VT G K)(3xn\t G Xn\t)(Vxt G Xt)F(xn\t,xt) T a (14)
holds.
A situation x0 G X is called quite K-acceptable if outcome F (x0) is quite K-acceptable one.
These optimality concepts are analogous to well known optimality concepts of games with payoff functions (see Moulin, 1981).
Now we consider connections between these optimality concepts.
Lemma 1. Nash K-equilibrium point is also a K-equilibrium point but converse is false.
Proof (of lemma). Let x0 = (x0)ieN be Nash K-equilibrium point then for any
P T
coalition T € K and any strategy xT € XT the condition F (x0 || xT) < F(x0) holds. Suppose F(x0 || xT) > F(x0). The system of conditions
{PT
F (x0 || xt ) < F(x0)
F(x0 || xt) > F(x0)
T
is false. Hence, F(x0 || xT) ^ F(x0). □
Thus, Nash K-equilibrium is K-equilibrium. But the converse is false. Indeed, consider
Example 1. Consider an antagonistic game G whose realization function F is given by Table 1 and preference relation for player 1 by Diagram 1; preference relation of player 2 is given by inverse order, K = {{1}, {2}} .
Table 1. Realization function
F
ti
t2
Si a b
82 c d
Situation (si,ti) is K-equilibrium. Since F (si,ti) = a and a||b, a||c (i.e. a and b is incomparable, a and c is incomparable) then (s1,t1) is not Nash K-equilibrium.
Remark 5. If all preference relations (pT)Te/C is linear then Nash K-equilibrium and K-equilibrium are equivalent.
Proposition 1. An objection of coalition T against situation x* is also a refutation of this situation.
Proof (of proposition). Let xT be an objection of coalition T against situation x*. Then according to definition of objection the strategy xT is an objection of coalition T against outcome F (x*), i.e. for any strategy of complementary coalition xN\ T €
T
XN\T the condition F (xT ,xN\T) > F (x*) holds.
Let us take xN\T = x*N\T as a strategy of complementary coalition, we have
T
F (xT ,x*N\T) > F (x*).
Since strategy xN\T is an arbitrary one then we get strategy xT is a refutation of this situation.
Fig. 1. Diagram 1
Corollary 1. Any K-equilibrium point is also K-acceptable.
We have to prove the more strong assertion.
Lemma 2. Any K-equilibrium point is also quite K-acceptable.
Proof (of lemma). Let x0 be K-equilibrium point. Suppose xN\T = x°N\T for all coalitions T € K. Then for any coalition T € K we have F (xN\T,xT) =
/ \ Pt
F ix°N\t,xT) = F(x0 || xT) ^ F(x0). Hence, x0 is quite K-acceptable. □
Lemma 3. Any quite K-acceptable outcome is K-acceptable.
The proof of Lemma 3 is obvious.
The main result of the part 3 is the following theorem.
Theorem 1. Consider introduced above coalitions optimality concepts: Nash K-equilibrium, K-equilibrium, quite K-acceptance, K-acceptance. Then each consequent condition is more weak than preceding, i.e.
Nash K-equilibrium ^ K-equilibrium ^ quite K-acceptance ^ K-acceptance.
The proof of Theorem 1 follows from Lemmas 1, 2, 3.
4. Coalition homomorphisms for games with preference relations
Let
G = ((Xi)ieN , A, F, (pi)ieN)
and
r = i(Yi)ieN , B, ^, (ai)ieN)
be two games with preference relations of the players N.
Any (n + 1)-system consisting of mappings f = (pi, . ..,pn, where for any
i = 1,...,n, : Xi ^ Yi and ^: A ^ B, is called a homomorphism from game G
into game r if for any i = 1,...,n and any ai, a2 € A the following two conditions
Pi °i
ai < a2 ^ ^(a\) < ^(a2), (15)
^(F (xi,..., xn)) = @(pi(xi), P2(x2),..., Pn(xn)) (16)
are satisfied.
A homomorphism f is said to be strict homomorphism if system of the conditions
Pi Ji
ai < a2 ^ ^(ai) <^(a2), (i = 1,...,n) (17)
ai P a2 ^ 4>(ai) ~ rfi(a2) (i = 1,...,n) (18)
holds instead of condition (15).
A homomorphism f is said to be regular homomorphism if the conditions
<Ji Pi
^(ai) < ^(a2) ^ ai < a2, (19)
^(ai) ~ ^(a2) ^ ^(ai) = ^2) (20)
hold.
A homomorphism f is said to be homomorphism ”onto”, if each pi (i = 1,...,n) is a mapping ”onto”.
Now we introduce a concept of coalition homomorphism.
For the first step, we need to fix some rule for concordance of preferences; recall that the preference relation for coalition T denoted by pT.
Definition 6. A homomorphism f is said to be:
— a coalition homomorphism if it preserves preference relations for all coalitions, i.e. for any coalition T C N the condition
T JT
ai < a2 ^ ^(ai) < ^fa) (21)
holds;
— a strict coalition homomorphism if for any coalition T C N the system of the conditions
{Pt Jt
ai < a2 ^ 4>(ai) < 'fifa), (22)
ai ~ a2 ^ ^(ai) JT 4>(a2)
is satisfied;
— a regular coalition homomorphism if for any coalition T C N the system of the conditions
{jt P t
fi>(ai) < ^(a2) ^ ai < a2, (23)
■fi(ai) JT ^(a2) ^ ^(ai) = ^(a2)
is satisfied.
It is easy to see that the following assertion is true.
Lemma 4. For Pareto concordance (and also for modified Pareto concordance),
any surjective homomorphism from G into r is a surjective coalition homomorphism.
Lemma 5. For Pareto concordance (and also for modified Pareto concordance), any strict homomorphism from G into r is a strict coalition homomorphism.
Proof (of lemma 5). We consider Pareto concordance for preferences as a concordance rule. Verify the conditions of system (22) for preference relation pT. According
Pt
to defition of Pareto concordance the condition ai < a2 is equivalent system
(Vi € T) ai < a2,
(3j € T) ai < a2.
Since strict homomorphism is homomorphism then from the first condition of
Ji
system it follows that (Vi € T) 0 (ai) < 0 (a2). Since homomorphism f is strict then
(3j € T) ^ (ai) < ^ (0,2) .
J t
From last two conditions we get 0 (ai) < 0 (a2).
Now according to definition of symmetric part of relation pT we have ai P^T a2 ^ (Vi € T) ai ~ a2. Since homomorphism f is strict then we get
(Vi € T) 0 (ai) ~ ^ (a2), i.e. 0 (ai) JT 0 (a2) . □
Now we consider modified Pareto concordance for preferences of players as a concordance rule.
Lemma 6. For modified Pareto concordance, any regular homomorphism from G into r is a regular coalition homomorphism.
Proof (of lemma 6). Verify the condition (23) for strict part of preference relation
aT. According to definition of modified Pareto concordance for preferences the con-
JT Ji
dition -0 (ai) < -0 (a2) is equivalent (Vi € T) 0 (ai) < 0 (a2). Since homomorphism
i T
f is regular then we have (Vi € T) ai < a2, i.e. ai < a2.
Verify the condition (23) for symmetric part of aT. According to definition of
modified Pareto concordance we have
■0 (ai) JT 0 (a2) ^ (Vi € T) 0 (ai) ~ 0 (a2) .
Since homomorphism f is regular then from the last condition it follows that
(Vi € T) 0 (ai) = 0 (a2), i.e. 0 (ai) = 0 (a2). □
5. The main results
The main result states a correspondence between sets of ^-acceptable outcomes and K-equilibrium situations of games which are in homomorphic relations under indicated types.
A homomorphism f is said to be covariant if /-image of any optimal solution in game G is an optimal solution in r.
A homomorphism f is said to be contrvariant if f-preimage of any optimal solution in game r is an optimal solution in G.
Theorem 2. For Nash K-equilibrium, any surjective homomorphism is covariant under Pareto concordance and under modified Pareto concordance also.
Proof (of theorem 2). We consider Pareto concordance for preferences as a concordance rule. Let x0 be Nash K-equilibrium point in game G. We have to prove that p (x0) is Nash K-equilibrium point in game r.
We fix arbitrary strategy yT € YT. Since f is homomorphism ”onto” then according to Lemma 4 we obtain (3xT € XT) pT (xT) = Vt ■ For any strat-
PT
egy xT the condition F(xT,x°N\t) < F(x0) holds. Hence, for strategy xT the
T
condition F(xT,x°N\t) < F(x0) is satisfied. Since f is homomorphism then 0 (F(xT, x0N\T) < 0 (F(x°)) ■ By condition (16): $ p (xT), Pn\t (x0N\T)) ^
$ (p (x°)) , Le. $ {vt, Pn\t (xN\t)) IT $ (p (x°)( ■
Since strategy yT € YT is arbitrary one then p (x0) is Nash K-equilibrium. □
Theorem 3. For K-equilibrium, any strict surjective homomorphism is contrvari-ant under Pareto concordance and under modified Pareto concordance also.
Proof (of theorem 3). Consider Pareto concordance for preferences as a concordance rule. Let y° be K-equilibrium point. We have to prove that situation x0 with p (x0) = y0 is K-equilibrium point.
Suppose x0 = (x°) n is not K-equilibrium then there exists coalition T € K
and strategy xT € XT such that F (xT, x°N\t^ > F (x0) ■ Since homomorphism
f is strict then according to Lemma 5 we get 0 ^F ^xT ,x°N\t^ > 0 (F (x0)) ■
According to condition (16) we obtain $ (pT (xT), pN\t (xNY^') > $ (p (x0)) ■
The last condition means $ (pT (xT), y°N\t^ > $ (y0) ■ Thus, strategy pT (xT)
is refutation of situation y0 by coalition T, which is contradictory with y0 is K-equilibrium point.
Hence, x0 is K-equilibrium point. □
Theorem 4. For K-acceptance, any strict surjective homomorphism is contrvari-ant under Pareto concordance and under modified Pareto concordance also.
Proof (of theorem 4). Consider Pareto concordance for preferences as a concordance rule. Let outcome b with 0 (a) = b be K-acceptable one in game r. Assume that
outcome a is not acceptable for all coalitions T €K, i.e. there exists such strategy xT € XT that for any strategy xN\t € XN\t the condition
PT
F(xt,xn\t) > a (24)
holds.
Let yN\t = (yj)jEN\t be arbitrary strategy of complementary coalition N \ T in game r. Since f is homomorphism ”onto” then according to Lemma 4 we have (3xN\t € Xn\t^J Pn\t (x*N\t^ = Vn\t■ By (24) the condition F(x‘T,x*N\t) > a
holds. According to Lemma 5 we get 0 (f(xi,x*NYe^ > 0 (a) ■ By (16)
we have 0 (f(xT,x*n\t)j = $ (pT (x'T) ,pN\t (x*N\t)) ■ Thus, the condition
$ (pT (xT) , yN\t) > 0 (a) is satisfied. Hence, strategy pT (xT) is objection of coalition T against outcome b which is contadictory with b is K-acceptable outcome.
Hence, outcome a is K-acceptable. □
Theorem 5. For K-equilibrium, any regular surjective homomorphism is covariant under modified Pareto concordance.
Proof (of theorem 5). Let x0 be K-equilibrium. We have to prove that situation p x0 is K-eq(uili)brium.
Suppose p (x0) is not K-equilibrium, i.e.
(3T € K) (3vt € Yt) $ (p (x0) ||vt) > $ (p (x0)) (25)
Since homomorphism f is surjective then accoeding to Lemma 4 we have (3xT € XT) pT (x’T) = yT■ Hence, the condition $ (pT (xT), pN\ t(xN\^) >
$ (p (x0)) holds. By (16) we get $ (pt (xT), Pn\t (x°N\^) = 0 (f (xT, x°N\^) ■ Thus, 0 (F (x0|xT)) > 0 (F (x^) ■ Because homomorphism f is regular then according to Lemma 6 we obtain F (x°||xT) > F (x^ ■ Thus, strategy xT is refutation of situation x(0 b)y coalition T, which is contradictory with x0 is K-equilibrium. Hence, p (x0) is K-equilibrium in game r. □
Appendix
Consider the example conserning of concordance rules.
Let G be a game of three players with set of outcomes A = {a, b, c, d, e}. Preference relations for each player are given by Diagrams 2,3,4.
Using Diagrams 2 - 4 we can define preference relations in the following form:
pi : a < b,b ~ c, c ~ d,b < e
P2 : a ~ b,b ~ c,c < d,e < d
P3 : a < c,b ~ c,c < d,b ~ e,d ~ e■
Then according to Pareto concordance (see 2.1) for coalition T = {1, 2} we have
PT PT
pT: a < b,b < c,c < d where strict part consists of two conditions a < b, c < d and symmetric part is b P'T c.
Pi e
a*
Fig. 2. Diagram 2
Fig. 3. Diagram 3
Fig. 4. Diagram 4
For T = {1, 3} a preference relation pT is defined by b < c,c < d,b < e where
Pt Pt pt
strict part is c < d,b < e and symmetric part is b ~ c.
Pt pt P\i
For T = {2, 3} relation pT is b < c,c < d,e < d where c < d, e < d,b t c.
For T = {1, 2, 3} relation pT is b < c,c < d where c < d,b TT c.
According to modified Pareto concordance (see 2.2) for coalition T = {1, 2} strict part pT is empty set and symmetric part consists of one condition b TT c.
P t
For T = {2, 3} strict part of preference relation pT is defined by c < d and symmetric part is b TT c.
Preference relation pT for coalition T = {1, 2, 3} in the game with majority rule
pT Pt pt pT pT
(see 2.3): a < b,b t c,c< d,b < e,e < d.
References
Savina, T. F. (2010). Hom,om,orphism,s and Congruence Relations for Games with Preference Relations. Contributions to game theory and management. Vol.III. Collected papers on the Third International Conference Game Theory and Management /Editors Leon A. Petrosyan, Nikolay A. Zenkevich.- SPb.: Graduate School of Management SPbU, pp. 387-398.
Rozen, V. V. (2009). Cooperative Games with Ordered Outcomes. Game Theory and Management. Collected abstracts of papers presented on the Third International Conference Game Theory and Management. SPb.: Graduate School of Management SPbU, pp. 221-222
Moulin, Herve. (1981). Theorie des jeux pour economie et la politique. Paris.
