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Applications of fixed point theory to extended Nash equilibriums of nonmonetized noncooperative games on posets
Fixed Point Theory and Applicationsvolume 2013, Article number: 235 (2013)
Abstract
We say that a noncooperative game is nonmonetized if the ranges of the utilities of the players are posets. In this paper, we examine some nonmonetized noncooperative games of which both the collection of strategies and the ranges of the utilities for the players are posets. Then we carry the concept of generalized Nash equilibriums of noncooperative games defined in (Li in J. Nonlinear Anal. Forum 18:111, 2013; Li and Park in Br. J. Econ. Manag. Trade 4(1), 2014) to extended Nash equilibriums of nonmonetized noncooperative games. By applying some fixed point theorems in posets and by using the orderpreserving property of mappings, we prove an existence theorem of extended Nash equilibriums for nonmonetized noncooperative games.
MSC:46B42, 47H10, 58J20, 91A06, 91A10.
1 Introduction
In economic theory, social science, military science, or other fields, the outcomes of some games may not be in a complete (totally) ordered set; that is, the utilities of the players may not be represented by realvalued functions, which is different from most noncooperative games in game theory. We give an example below to more precisely demonstrate our arguments.
Tankerjet example Suppose that a military project industry produces a certain type of tankers and a certain type of fighter planes. An arbitrary considered outcome is a set of tankers and fighter planes produced by this industry. Let U be the collection of all possible outcomes. Assume that the combat effectiveness of the tankers and the fighter planes is not comparable. Denote an arbitrary element in U by $(x,y)$ where x, y are the numbers of tankers and fighter planes in this outcome, respectively. Then, based on the incomparable capability of the outcomes, the nonrational preference relation on U is defined by
This preference relation ⪰ on U is not a complete order, which is a partial order on U. In case the budget of this military industry is limited (this is always true in the real world), the decision makers of this industry would seek the war production plan to optimize the fighting capability under some given warfare environment.
It is clear that such an optimization problem is not a normal optimization problem (with realvalued functions). As a matter of fact, these optimization ideas, introduced and studied by Giannessi [1] in 1980, are called vector optimization problems and vector variational inequalities, where the outcome space is in a finitedimensional vector space. Since then, the theory as well as algorithms of vector variational inequalities and vector optimization have been extensively studied. See, e.g., [2–9] and the references therein.
Understanding the existence of optimization problems with respect to a partial order on a set of objects U is of considerable interest for Li [10], Li and Park [11] to introduce the concepts of the generalized Nash equilibrium of nonmonetized noncooperative games, in which the utilities of the players may not be realvalued; that is, the payoffs of the players may not be represented by realvalued functions. Now we recall the concept of generalized Nash equilibrium of nonmonetized noncooperative games from [10, 11].
In an nperson nonmonetized noncooperative game $\mathrm{\Gamma}=(N,S,P,U)$, where $(U,{\u2ab0}^{U})$ is a lattice, a selection of strategies $({\overline{x}}_{1},{\overline{x}}_{2},\dots ,{\overline{x}}_{n})\in {S}_{1}\times {S}_{2}\times \cdots \times {S}_{n}$ is called a generalized Nash equilibrium of this game if the following order inequality holds:
Existence theorems of generalized Nash equilibriums of some nonmonetized noncooperative games are provided in [10, 11]. In [10] the space of outcomes is a Banach lattice, and in [11] it is just a lattice, which is not required to be equipped with neither topological structure nor algebraic structure.
In this paper, we extend the concept of generalized Nash equilibrium to broader classes as follows: a selection of strategies $({\overline{x}}_{1},{\overline{x}}_{2},\dots ,{\overline{x}}_{n})\in {S}_{1}\times {S}_{2}\times \cdots \times {S}_{n}$ is called an extended Nash equilibrium of a nonmonetized noncooperative game $\mathrm{\Gamma}=(N,S,P,U)$ if the following order inequality holds:
It is clear that in both pure mathematical theory and real applications the order inequality (ii) is more general than the order inequality (i). In Section 3, we prove an existence theorem of extended Nash equilibriums of some nonmonetized noncooperative games.
2 Preliminaries
For the readers’ convenience, in this section, we recall some concepts and properties of posets and lattices (see, e.g., [7, 12–15]). Here we closely follow the notations from [7] and [15].
Let $P=(P,\u2ab0)$ be a poset. An element u of P is called an upper bound of a subset A of P if $x\u2aafu$ for each $x\in A$. If $u\in A$, then u is called the greatest element of A and is denoted by $u=maxA$. If the set of all upper bounds of A has the smallest element, we call it the supremum of A and denote it by supA or ∨A. An element y is called a maximal element of A if $y\in A$ and if $z\in A$ and $y\u2aafz$ imply that $y=z$. A lower bound of A, the smallest element minA of A, the infimum of A, $infA=\wedge A$ and a minimal element of A can be defined similarly.
A poset $P=(P,\u2ab0)$ is called a lattice if $inf\{x,y\}$ and $sup\{x,y\}$ exist for all $x,y\in P$. Define $inf\{x,y\}=x\wedge y$ and $sup\{x,y\}=x\vee y$. A subset C of a poset $P=(P,\u2ab0)$ is called a chain if $x\u2aafy$ or $y\u2aafx$ for all $x,y\in C$.
Definition 2.1 A poset $P=(P,\u2ab0)$ is said to be

(i)
inductive if every chain in P has an upper bound in P;

(ii)
inversely inductive if every chain in P has a lower bound in P;

(iii)
biinductive whenever it is both inductive and inversely inductive.
Definition 2.2 A poset $P=(P,\u2ab0)$ is said to be

(i)
strongly inductive whenever for every chain C in P, the supremum of C, supC, exists in P;

(ii)
strongly inversely inductive whenever for every chain C in P, the infimum of C, infC, exists in P;

(iii)
strongly biinductive whenever it is both strongly inductive and strongly inversely inductive.
Extension of Zorn’s lemma Every biinductive poset has both maximal and minimal elements.
Let $P=(P,\u2ab0)$ be a poset. For any $z,w\in P$, we define the order intervals:
Given posets $(X,{\u2ab0}^{X})$ and $(U,{\u2ab0}^{U})$, we say that a mapping $F:X\to {2}^{U}\mathrm{\setminus}\mathrm{\varnothing}$ is order increasing upward if $x{\u2aaf}^{X}y$ in X and $z\in F(x)$ imply that $[z)\cap F(y)$ is nonempty, that is, if $x{\u2aaf}^{X}y$ in X and $z\in F(x)$, then there is $w\in F(y)$ such that $z{\u2aaf}^{U}w$. The mapping F is order increasing downward if $x{\u2aaf}^{X}y$ in X and $w\in F(y)$ imply that $(w]\cap F(x)$ is nonempty. F is said to be order increasing whenever F is both order increasing upward and downward.
As a special case, a singlevalued mapping F from a poset $(X,{\u2ab0}^{X})$ to the poset $(U,{\u2ab0}^{U})$ is said to be order increasing whenever $x{\u2aaf}^{X}y$ implies $F(x){\u2aaf}^{U}F(x)$. An increasing mapping $F:X\to U$ is said to be strictly order increasing whenever $x{\prec}^{X}y$ implies $F(x){\prec}^{U}F(x)$.
A nonempty subset A of a subset Y of a poset $P=(P,\u2ab0)$ is said to be order compact upward in Y if for every chain C of Y that has a supremum in P, the intersection $\bigcap \{[y)\cap A:y\in C\}$ is nonempty whenever $[y)\cap A$ is nonempty for every $y\in C$. The set A is order compact downward in Y if for every chain C of Y that has the infimum in P, the intersection $\bigcap \{(y]\cap A:y\in C\}$ is nonempty whenever $(y]\cap A$ is nonempty for every $y\in C$. If A is both order compact upward and order compact downward in Y, then A is said to be order compact in Y.
Let A be a subset of a poset $P=(P,\u2ab0)$. An element $c\in P$ is called a supcenter of A in P if $c\vee x$ exists in P for each $x\in A$. If $c\wedge x$ exists in P for each $x\in A$, then c is called an infcenter of A in P. If c is both a supcenter and an infcenter of A in P, then c is called an order center of A in P. In particular, if $A=P$, then c is simply called a supcenter or an infcenter of P, respectively.
Let A be a nonempty subset of a poset $P=(P,\u2ab0)$. The set $ocl(A)$ is the collection of all possible supremums and infimums of chains of A, which is called the order closure of A. If $A=ocl(A)$, then A is said to be order closed.
Remark 2.3 Every nonempty strongly biinductive subset of a poset $P=(P,\u2ab0)$ is order closed.
Now we recall a fixed point theorem on posets from [16]. It will be used in the proof of the existence of extended Nash equilibrium of nonmonetized noncooperative games in Section 3.
Theorem 2.4 [[16], Theorem 2.12]
Let $P=(P,\u2ab0)$ be a poset. Assume that a setvalued mapping $F:P\to {2}^{P}\mathrm{\setminus}\mathrm{\varnothing}$ is order increasing, and that its values are order compact in $F(P)$. If chains of $F(P)$ have supremums and infimums (in P), and if $ocl(F(P))$ has a supcenter or an infcenter in P, then F has minimal and maximal fixed points.
3 Nonmonetized noncooperative games on posets
In the section of Introduction, we described the motivations to extend the noncooperative games and the concept of generalized Nash equilibrium to nonmonetized noncooperative games. In this section, we give some definitions for these extensions, which are the generalized notions of Nash equilibriums defined and studied in games theory (see, e.g., [16–21]).
Definition 3.1 Let n be a positive integer greater than 1. An nperson nonmonetized noncooperative game consists of the following elements:

(1)
the set of n players, which is denoted by $N=\{1,2,3,\dots ,n\}$;

(2)
the collection of n strategy sets $S=\{{S}_{1},{S}_{2},\dots ,{S}_{n}\}$, for the n players respectively, which is also written as $S={S}_{1}\times {S}_{2}\times \cdots \times {S}_{n}$;

(3)
the outcome space $(U;{\u2ab0}^{U})$ that is a poset;

(4)
the n utilities functions (payoff mappings) ${P}_{1},{P}_{2},\dots ,{P}_{n}$, where ${P}_{i}$ is the utility function for player i that is a mapping from ${S}_{1}\times {S}_{2}\times \cdots \times {S}_{n}$ to the poset $(U;{\u2ab0}^{U})$ for $i=1,2,3,\dots ,n$. We define $P=\{{P}_{1},{P}_{2},\dots ,{P}_{n}\}$.
This game is denoted by $\mathrm{\Gamma}=(N,S,P,U)$.
In an nperson nonmonetized noncooperative game $\mathrm{\Gamma}=(N,S,P,U)$, when all the n players $1,2,3,\dots ,n$ simultaneously and independently choose their own strategies ${x}_{1},{x}_{2},\dots ,{x}_{n}$ to play, where ${x}_{i}\in {S}_{i}$, for $i=1,2,3,\dots ,n$, then player i will receive his utility (payoff) ${P}_{i}({x}_{1},{x}_{2},\dots ,{x}_{n})\in U$. For any $x=({x}_{1},{x}_{2},\dots ,{x}_{n})\in S$, and for every given $i=1,2,3,\dots ,n$, as usual, we define
Then ${x}_{i1}\in {S}_{i1}$ and x can be simply written as $x=({x}_{i},{x}_{i1})$. Moreover, we define
Now we extend the concept of Nash equilibrium of noncooperative games and generalized Nash equilibrium of nonmonetized noncooperative games to the extended Nash equilibrium of nonmonetized noncooperative games.
Definition 3.2 In an nperson nonmonetized noncooperative game $\mathrm{\Gamma}=(N,S,P,U)$, a selection of strategies $({\tilde{x}}_{1},{\tilde{x}}_{2},\dots ,{\tilde{x}}_{n})\in {S}_{1}\times {S}_{2}\times \cdots \times {S}_{n}$ is called an extended Nash equilibrium of this game if, for every $i=1,2,3,\dots ,n$, the following order inequality holds:
Lemma 3.3 Let $({S}_{i},{\u2ab0}_{i})$ be a poset for every $i=1,2,\dots ,n$. Let $S={S}_{1}\times {S}_{2}\times \cdots \times {S}_{n}$ be the Cartesian product space of ${S}_{1},{S}_{2},\dots ,{S}_{n}$, and let ${\u2ab0}^{S}$ be the coordinate partial order on S induced by the partial orders ${\u2ab0}_{i}$. That is, for any $x,y\in S$ with $x=({x}_{1},{x}_{2},\dots ,{x}_{n})$ and $y=({y}_{1},{y}_{2},\dots ,{y}_{n})$,
Then $(S,{\u2ab0}^{S})$ is a poset. Furthermore, if every $({S}_{i},{\u2ab0}_{i})$ is (strongly) inductive, then $(S,{\u2ab0}^{S})$ is also (strongly) inductive. If every $({S}_{i},{\u2ab0}_{i})$ is (strongly) biinductive, then $(S,{\u2ab0}^{S})$ is also (strongly) biinductive.
Proof The proof is straightforward and is omitted here. □
The following theorem is the main result of this paper. It provides some conditions for the existence of the extended Nash equilibrium of nonmonetized noncooperative games.
Theorem 3.4 Let $\mathrm{\Gamma}=(N,S,P,U)$ be an nperson nonmonetized noncooperative game. If for every player $i=1,2,3,\dots ,n$, his strategy set $({S}_{i},{\u2ab0}_{i})$ satisfies:

(i)
$({S}_{i},{\u2ab0}_{i})$ is a strongly inversely inductive poset;

(ii)
$({S}_{i},{\u2ab0}_{i})$ has a supcenter (or an infcenter simultaneously for all i);
and his utility function ${P}_{i}$ satisfies the following conditions:

(1)
${P}_{i}:S\to U$ is (singlevalued) order increasing with respect to the product order ${\u2ab0}^{S}$;

(2)
for any fixed ${x}_{i}\in {S}_{i}$, ${P}_{i}({S}_{i},{x}_{i})$ is an inductive subset of $(U,{\u2ab0}^{U})$;

(3)
for any fixed ${x}_{i}\in {S}_{i}$, and for any $u\in {P}_{i}({S}_{i},{x}_{i})$, the inverse image $\{{z}_{i}\in {S}_{i}:{P}_{i}({z}_{i},{x}_{i})=u\}$ is a strongly biinductive subset of ${S}_{i}$;

(4)
for any $({x}_{i},{x}_{i}),({y}_{i},{y}_{i})\in S$ satisfying $({x}_{i},{x}_{i}){\u2aaf}^{S}({y}_{i},{y}_{i})$, the maximal elements have the following monotone properties:

(a)
if ${P}_{i}({z}_{i},{x}_{i})$ is a maximal element of ${P}_{i}({S}_{i},{x}_{i})$, then there is ${w}_{i}\in {S}_{i}$ with ${z}_{i}{\u2aaf}_{i}{w}_{i}$ such that ${P}_{i}({w}_{i},{y}_{i})$ is a maximal element of ${P}_{i}({S}_{i},{y}_{i})$;

(b)
if ${P}_{i}({w}_{i},{y}_{i})$ is a maximal element of ${P}_{i}({S}_{i},{y}_{i})$, then there is ${z}_{i}\in {S}_{i}$ with ${z}_{i}{\u2aaf}_{i}{w}_{i}$ such that ${P}_{i}({z}_{i},{x}_{i})$ is a maximal element of ${P}_{i}({S}_{i},{x}_{i})$.

(a)
Then this nonmonetized noncooperative game Γ has an extended Nash equilibrium. Furthermore, this nonmonetized noncooperative game Γ has minimal and maximal (with respect to the product lattice order ${\u2ab0}^{S}$) extended Nash equilibriums.
Proof For every $i=1,2,\dots ,n$, since $({S}_{i},{\u2ab0}_{i})$ is a poset, then from Lemma 3.3, $(S,{\u2ab0}^{S})$ is also a poset equipped with the product order ${\u2ab0}^{S}$. For every fixed $i=1,2,3,\dots ,n$, define a setvalued mapping ${T}_{i}:S\to {2}^{{S}_{i}}\mathrm{\setminus}\mathrm{\varnothing}$ by
From condition (2) of this theorem, for every fixed element ${x}_{i}\in {S}_{i}$, ${P}_{i}({S}_{i},{x}_{i})$ is an inductive subset of U. Then, applying the extension of Zorn’s lemma, the range ${P}_{i}({S}_{i},{x}_{i})$ has a maximal element; and hence ${T}_{i}(x)$ is a nonempty subset of ${S}_{i}$. Furthermore, we claim that
In fact, pick an arbitrary chain ${C}_{i}=\{{z}_{i}:{z}_{i}\in {C}_{i}\}$ in ${T}_{i}(x)\subseteq {S}_{i}$. Then the order increasing property of ${P}_{i}$ implies that $\{{P}_{i}({z}_{i},{x}_{i}):{z}_{i}\in {C}_{i}\}$ must be a chain in ${P}_{i}({S}_{i},{x}_{i})$. On the other hand, from (1), $\{{P}_{i}({z}_{i},{x}_{i}):{z}_{i}\in {C}_{i}\}$ is a collection of some maximal elements of ${P}_{i}({S}_{i},{x}_{i})$. Its chain property implies that $\{{P}_{i}({z}_{i},{x}_{i}):{z}_{i}\in {C}_{i}\}$ must be a singleton. So, we can assume ${P}_{i}({z}_{i},{x}_{i})=u$ for all ${z}_{i}\in {C}_{i}$ and for some $u\in (U,{\u2ab0}^{U})$. Then the set (chain) ${C}_{i}$ is a subset of the inverse image $\{{t}_{i}\in {S}_{i}:{P}_{i}({t}_{i},{x}_{i})=u\}$; that is,
From condition (3), $\{{t}_{i}\in {S}_{i}:{P}_{i}({t}_{i},{x}_{i})=u\}$ is a strongly biinductive subset of ${S}_{i}$. It implies that the chain ${C}_{i}$ has both supremum and infimum in the inverse image $\{{t}_{i}\in {S}_{i}:{P}_{i}({t}_{i},{x}_{i})=u\}$, which is contained in ${T}_{i}(x)$. They clearly are the supremum and infimum of this chain ${C}_{i}$ in ${T}_{i}(x)$, respectively. It proves claim (2).
Define the product mapping $T:S\to {2}^{S}\mathrm{\setminus}\mathrm{\varnothing}$ of the setvalued mappings ${T}_{1},{T}_{2},\dots ,\text{and}{T}_{n}$ by
From (2) and applying Lemma 3.3, we obtain that $T(x)$ is a nonempty strongly biinductive subset of $(S,{\u2ab0}^{U})$ for every $x=({x}_{1},{x}_{2},\dots ,{x}_{n})\in S$.
In order to apply Theorem 2.12 in [16] to show the existence of a fixed point for this mapping $T:S\to {2}^{S}\mathrm{\setminus}\mathrm{\varnothing}$, we have to show that the setvalued mapping T is increasing, that its values are order compact in $T(S)$, that chains of $T(S)$ have supremums and infimums, and that $ocl(T(S))$ has a supcenter or an infcenter in S.
Now we show that $T:S\to S$ is an order increasing setvalued mapping with respect to the product lattice order ${\u2ab0}^{S}$. At first, we prove the order increasing upward property for T. For any given $x,y\in S$, with $x{\u2aaf}^{S}y$, and for any given $z\in T(x)$, we need to show that there is $w\in T(y)$ such that $z{\u2aaf}^{S}w$. We write $z=({z}_{1},{z}_{2},\dots ,{z}_{n})$. The definition of the product mapping T and the hypothesis $z\in T(x)$ imply ${z}_{i}\in {T}_{i}(x)$; that is,
Applying part (a) in condition (4) of this theorem, and from $({x}_{i},{x}_{i}){\u2aaf}^{S}({y}_{i},{y}_{i})$, there is ${w}_{i}\in {S}_{i}$ such that
and ${P}_{i}({w}_{i},{y}_{i})$ is a maximal element of ${P}_{i}({S}_{i},{y}_{i})$; that is,
Let $w=({w}_{1},{w}_{2},\dots ,{w}_{n})$. By combining (3) and (4), we obtain $z{\u2aaf}^{S}w$ and $w\in T(y)$. It proves that T is an order increasing upward mapping on S.
Secondly, we prove the order increasing downward property for T. For any given $x,y\in S$, with $x{\u2aaf}^{S}y$, and for any $w\in T(y)$, we need to show that there is $z\in T(x)$ such that $z{\u2aaf}^{S}w$. Write $w=({w}_{1},{w}_{2},\dots ,{w}_{n})$. The hypothesis $w\in T(y)$ implies ${w}_{i}\in {T}_{i}(y)$ for every $i=1,2,\dots ,n$. That is, ${P}_{i}({w}_{i},{y}_{i})$ is a maximal element of ${P}_{i}({S}_{i},{y}_{i})$. Applying part (b) of condition (4) of this theorem, and from $({w}_{i},{x}_{i}){\u2aaf}^{S}({w}_{i},{y}_{i})$ that is implied by the condition $x{\u2aaf}^{S}y$, we obtain that ${P}_{i}({w}_{i},{x}_{i})$ is a maximal element of ${P}_{i}({S}_{i},{x}_{i})$; that is, ${w}_{i}\in {T}_{i}(x)$. Notice ${w}_{i}{\u2aaf}_{i}{w}_{i}$ for every $i=1,2,\dots ,n$. It implies $w\in T(x)$; and hence T is order increasing downward. Thus T is an order increasing setvalued mapping.
Next we show that $T:S\to S$ has order compact values in $T(S)$. That is, we need to show that for every $x\in S$, $T(x)$ is both order compact upward and order compact downward in $T(S)$. At first we show the order compact upward property of the values of T. Take an arbitrary chain C in $T(S)$ that has a supremum in S, which satisfies that
(Note that the condition that the given chain C in $T(S)$ has a supremum in S is not used in the following proof of the order compact upward property of the values of T.) Define $y=({y}_{1},{y}_{2},\dots ,{y}_{n})$. Then
Since C is a chain in $T(S)$ (which has a supremum in S), the set $\{{y}_{i}:y\in C\}$ is a chain in ${T}_{i}(S)$ (and it also has a supremum in ${S}_{i}$). Hypothesis (5) and equality (6) imply
From inequality (7) we obtain that for every fixed $i=1,2,3,\dots ,n$ and for every fixed $y\in C$, there is $\mu ({y}_{i})\in {S}_{i}$ satisfying
(Where μ can be considered as a function of two variables y and i defined on $C\times N$ satisfying $\mu :\{y\}\times N\to S$ for every fixed $y\in C$ and $\mu :C\times \{i\}\to {S}_{i}$ for every fixed $i\in N$.) So, for every fixed i, when y varies in the chain C, from (8) we obtain a subset $\{\mu ({y}_{i}):y\in C\}\subseteq {T}_{i}(x)\subseteq {S}_{i}$ that is a chain in ${T}_{i}(x)$. From (2), since ${T}_{i}(x)$ is strongly biinductive and order closed, the chain $\{\mu ({y}_{i}):y\in C\}$ in ${T}_{i}(x)$ has its supremum and infimum in ${T}_{i}(x)$. Let the supremum of $\{\mu ({y}_{i}):y\in C\}$ be denoted by
(Note: After taking the supremum operator with respect to $y\in C$, ${q}_{i}$ only depends on i, and it does not depend on y in C.) By combining (8) and (9), for every fixed $i=1,2,3,\dots ,n$, we have
which implies
Let $q=({q}_{1},{q}_{2},\dots ,{q}_{n})$. From (6) and (10), we obtain $q\in \bigcap \{[y)\cap T(x):y\in C\}$. That is,
Hence, $T(x)$ is order compact upward in $T(S)$.
By the strongly biinductive property (2), very similarly to the above proof, we can show that for every $x\in S$, $T(x)$ is order compact downward in $T(S)$. Hence $T(x)$ is order compact in $T(S)$.
From (2), we see that for every $x\in S$, $T(x)$ is a strongly biinductive subset of S. Then we show that $T(S)$ is a strongly biinductive subset of S. It is equivalent to that every chain of $T(S)$ has both supremum and infimum in S (not necessarily to be in $T(S)$). Suppose that
is a chain in $T(S)$ with index set Λ. Take a fixed $\gamma \in \mathrm{\Lambda}$; that is, a fixed ${z}^{\gamma}\in C$, there is ${x}^{\gamma}=({x}_{1}^{\gamma},{x}_{2}^{\gamma},\dots ,{x}_{n}^{\gamma})\in S$ such that ${z}^{\gamma}\in T({x}^{\gamma})={T}_{1}({x}^{\gamma})\times {T}_{2}({x}^{\gamma})\times \cdots \times {T}_{n}({x}^{\gamma})$. Then, for $i=1,2,3,\dots ,n$, it implies ${z}_{i}^{\gamma}\in {T}_{i}({x}^{\gamma})$; that is,
For any ${z}^{\lambda}\in C$ with ${z}^{\lambda}{\u2ab0}^{S}{z}^{\gamma}$, we have ${z}_{i}^{\lambda}{\u2ab0}_{i}{z}_{i}^{\gamma}$, that implies $({z}_{i}^{\lambda},{x}_{i}^{\gamma}){\u2ab0}^{S}({z}_{i}^{\gamma},{x}_{i}^{\gamma})$ for $i=1,2,3,\dots ,n$. From the order increasing property of ${P}_{i}$, we get
Combining (11) and (12), and from ${z}_{i}^{\lambda}\in {S}_{i}$, we obtain ${P}_{i}({z}_{i}^{\lambda},{x}_{i}^{\gamma})={P}_{i}({z}_{i}^{\gamma},{x}_{i}^{\gamma})$, which is a maximal element of ${P}_{i}({S}_{i},{x}_{i}^{\gamma})$ which implies that ${z}^{\lambda}\in C$ with ${z}^{\lambda}{\u2ab0}^{S}{z}^{\gamma}$. Then we have ${z}_{i}^{\lambda}\in {T}_{i}({x}^{\gamma})$; and therefore
Since C is a chain in $T(S)$, it clearly implies that $\{{z}_{i}^{\lambda}:{z}^{\lambda}\in C\text{and}{z}^{\lambda}{\u2ab0}^{S}{z}^{\gamma}\}$ is a chain in ${T}_{i}({x}^{\gamma})$. From (2), this chain $\{{z}_{i}^{\lambda}:{z}^{\lambda}\in C\text{and}{z}^{\lambda}{\u2ab0}^{S}{z}^{\gamma}\}$ in ${T}_{i}({x}^{\gamma})$ has its supremum in ${T}_{i}({x}^{\gamma})$. Hence $\bigvee \{{z}_{i}^{\lambda}:{z}^{\lambda}\in C\text{and}{z}^{\lambda}{\u2ab0}^{S}{z}^{\gamma}\}$ exists and it is in ${T}_{i}({x}^{\gamma})$, which is denoted by
Let $\nu =({\nu}_{1},{\nu}_{2},\dots ,{\nu}_{n})$. Then the first part of (13) implies $\nu =\bigvee \{{z}^{\lambda}\in C:{z}^{\lambda}{\u2ab0}^{S}{z}^{\gamma}\}$. Since C is a chain, and notice ${z}^{\lambda}\in C$, it yields that
Hence ν is the supremum of this chain C in S. (Furthermore, the second part of (13) implies that $\nu \in T({x}^{\gamma})$, that is, the supremum of this chain C in $T(S)$ is in $T(S)$.)
On the other hand, for the given chain $C=\{{z}^{\lambda}=({z}_{1}^{\lambda},{z}_{2}^{\lambda},\dots ,{z}_{n}^{\lambda}):\lambda \in \mathrm{\Lambda}\}\subseteq T(S)$, we recognize that $\{{z}_{i}^{\lambda}:{z}^{\lambda}\in C\}$ is a chain in ${S}_{i}$. The inversely inductive property of ${S}_{i}$ given in condition (i) in this theorem guarantees the existence of the infimum of the chain $\{{z}_{i}^{\lambda}:{z}^{\lambda}\in C\}$ in ${S}_{i}$. Then
exists and is in ${S}_{i}$. Let $b=({b}_{1},{b}_{2},\dots ,{b}_{n})$. From the above definition, b is the infimum of this chain C in S. Hence, for any arbitrary chain C in $T(S)$, both supC and infC exist in S.
Finally, from condition (ii) of this theorem, suppose that ${S}_{i}$ has an infcenter for every $i=1,2,\dots ,n$. Let ${c}_{i}$ be an infcenter of ${S}_{i}$ for every $i=1,2,\dots ,n$, and let $c=({c}_{1},{c}_{2},\dots ,{c}_{n})$. It is clear that c is an infcenter of S. So, c is an infcenter of $ocl(T(S))$ in S. Thus the mapping $T:S\to {2}^{S}\mathrm{\setminus}\mathrm{\varnothing}$ satisfies all the conditions of Theorem 2.4. Hence T has a fixed point $\tilde{x}=({\tilde{x}}_{1},{\tilde{x}}_{2},\dots ,{\tilde{x}}_{n})$ which satisfies $\tilde{x}\in T(\tilde{x})$. It implies ${\tilde{x}}_{i}\in {T}_{i}(\tilde{x})$; that is,
It is equivalent to that for every $i=1,2,3,\dots ,n$, we have
which shows that $\tilde{x}=({\tilde{x}}_{1},{\tilde{x}}_{2},\dots ,{\tilde{x}}_{n})$ is an extended Nash equilibrium of this game. Furthermore, from Theorem 2.4, the mapping T has minimal and maximal fixed points. Based on the above argument, they are the minimal and maximal (with respect to the product lattice order ${\u2ab0}^{S}$) extended Nash equilibriums for this game. This completes the proof of this theorem. □
4 Nonmonetized noncooperative games on lattices
In this section, we consider a special case of nonmonetized noncooperative games, defined in Definition 3.1 in the last section, for the outcome space and strategy sets to be lattices. Then we examine that the generalized Nash equilibriums of nonmonetized noncooperative games defined in [10, 11] are the special cases of the extended Nash equilibriums defined in Definition 3.2.
Definition 4.1 [[11], Definition 3.2]
In an nperson nonmonetized noncooperative game $\mathrm{\Gamma}=(N,S,P,U)$, with $(U,{\u2ab0}^{U})$ to be a lattice, a selection of strategies $({\overline{x}}_{1},{\overline{x}}_{2},\dots ,{\overline{x}}_{n})\in {S}_{1}\times {S}_{2}\times \cdots \times {S}_{n}$ is called a generalized Nash equilibrium of this game if the following order inequality holds
for all ${x}_{i}\in {S}_{i}$ and for every $i=1,2,3,\dots ,n$.
It is clear that any generalized Nash equilibrium of a nonmonetized noncooperative game is an extended Nash equilibrium of this game. The converse may not be true. We show below that Theorem 3.4 in [11] about the existence of generalized Nash equilibrium can be obtained as a corollary of Theorem 3.4 in this paper.
Corollary 4.2 [[11], Theorem 3.4]
Let $\mathrm{\Gamma}=(N,S,P,U)$ be an nperson nonmonetized noncooperative game, with $(U,{\u2ab0}^{U})$ to be a lattice. If for every player $i=1,2,3,\dots ,n$, his strategy set $({S}_{i},{\u2ab0}_{i})$ is a strongly inversely inductive lattice and his utility function ${P}_{i}$ satisfies the following conditions:

(1)
${P}_{i}:S\to U$ is (singlevalued) order increasing with respect to the product order ${\u2ab0}^{S}$;

(2)
for any fixed ${x}_{i}\in {S}_{i}$, ${P}_{i}({S}_{i},{x}_{i})$ is an order bounded and Dedekind complete subset of $(U,{\u2ab0}^{U})$;

(3)
for any fixed ${x}_{i}\in {S}_{i}$, and for any $u\in {P}_{i}({S}_{i},{x}_{i})$, the inverse image $\{{z}_{i}\in {S}_{i}:{P}_{i}({z}_{i},{x}_{i})=u\}$ is an order bounded and Dedekind complete subset of ${S}_{i}$;

(4)
if $({w}_{i},{x}_{i}){\u2aaf}^{S}({w}_{i},{y}_{i})$, then for any $u\in {P}_{i}({S}_{i},{x}_{i})\cap ({P}_{i}({w}_{i},{y}_{i})]$, there is ${z}_{i}\in {S}_{i}$ such that ${z}_{i}{\u2aaf}_{i}{w}_{i}$ and ${P}_{i}({z}_{i},{x}_{i})=u$.
Then this nonmonetized noncooperative game Γ has a generalized Nash equilibrium. Furthermore, Γ has minimal and maximal (with respect to the product lattice order ${\u2ab0}^{S}$) generalized Nash equilibriums.
Proof One can see that any order bounded Dedekind complete subset of a lattice is a strongly biinductive subset of this lattice. Hence conditions (2) and (3) in this corollary imply conditions (2) and (3) in Theorem 3.4. Next, we show that conditions (1)(4) in this corollary imply condition (4) in Theorem 3.4. As a matter of fact, under condition (2) in this corollary, we have that ${P}_{i}({w}_{i},{y}_{i})$ is a maximal element of ${P}_{i}({S}_{i},{y}_{i})$ if and only if ${P}_{i}({w}_{i},{y}_{i})={\bigvee}_{{t}_{i}\in {S}_{i}}{P}_{i}({t}_{i},{y}_{i})$. Suppose $({w}_{i},{x}_{i}){\u2aaf}^{S}({w}_{i},{y}_{i})$. From condition (1), ${P}_{i}({t}_{i},{x}_{i}){\u2aaf}^{U}{P}_{i}({t}_{i},{y}_{i})$ for all ${t}_{i}\in {S}_{i}$. It implies
Since ${P}_{i}({S}_{i},{x}_{i})$ is order bounded Dedekind complete, we have
Combining (14) and (15) yields
From condition (4) in this corollary, there is ${z}_{i}\in {S}_{i}$ such that ${z}_{i}{\u2aaf}_{i}{w}_{i}$ and ${P}_{i}({z}_{i},{x}_{i})={\bigvee}_{{t}_{i}\in {S}_{i}}{P}_{i}({t}_{i},{x}_{i})$. By applying the order increasing property of ${P}_{i}$, we have ${P}_{i}({w}_{i},{x}_{i}){\u2ab0}^{S}{P}_{i}({z}_{i},{x}_{i})={\bigvee}_{{t}_{i}\in {S}_{i}}{P}_{i}({t}_{i},{x}_{i})$. We then obtain
So, ${P}_{i}({w}_{i},{x}_{i})$ is also the supremum of ${P}_{i}({S}_{i},{x}_{i})$ in ${P}_{i}({S}_{i},{x}_{i})$, which is the unique maximal element of ${P}_{i}({S}_{i},{x}_{i})$. Taking ${z}_{i}={w}_{i}$, then part (b) of condition (4) in Theorem 3.4 holds.
To show that part (a) of condition (4) in Theorem 3.4 holds, for any given $x,y\in S$, with $x{\u2aaf}^{S}y$, and for any given ${z}_{i}\in {S}_{i}$ such that ${P}_{i}({z}_{i},{x}_{i})$ is the supremum of ${P}_{i}({S}_{i},{x}_{i})$ in ${P}_{i}({S}_{i},{x}_{i})$, which is the unique maximal element of ${P}_{i}({S}_{i},{x}_{i})$; that is,
For any fixed ${t}_{i}\in {S}_{i}$, the condition $x{\u2aaf}^{S}y$ clearly implies $({t}_{i},{x}_{i}){\u2aaf}^{S}({t}_{i},{y}_{i})$. Then, from condition (1) of this corollary, we obtain
which implies
From condition (2) of this corollary again, the range ${P}_{i}({S}_{i},{y}_{i})$ is an order bounded and Dedekind complete subset of U. Then there is ${a}_{i}\in {S}_{i}$ such that
Let ${w}_{i}={z}_{i}\vee {a}_{i}\in {S}_{i}$. Then ${w}_{i}$ satisfies
The second order inequality above implies $({w}_{i},{y}_{i}){\u2ab0}^{S}({a}_{i},{y}_{i})$. The orderincreasing property of ${P}_{i}:S\to U$ implies
Since ${w}_{i}={z}_{i}\vee {a}_{i}\in {S}_{i}$, from the above order inequality, we get ${P}_{i}({w}_{i},{y}_{i})={\bigvee}_{{t}_{i}\in {S}_{i}}{P}_{i}({t}_{i},{y}_{i})$. Hence ${P}_{i}({w}_{i},{y}_{i})$ is the supremum of ${P}_{i}({S}_{i},{y}_{i})$ in ${P}_{i}({S}_{i},{y}_{i})$, which is the unique maximal element of ${P}_{i}({S}_{i},{y}_{i})$. It shows that part (a) of condition (4) in Theorem 3.4 holds. Hence condition (4) in Theorem 3.4 is satisfied.
Condition (ii) for the set of strategies ${S}_{i}$ in Theorem 3.4 is satisfied. Every element of ${S}_{i}$ is both a supcenter and an infcenter of ${S}_{i}$. It is because $({S}_{i},{\u2ab0}_{i})$ is a strongly inversely inductive lattice. So, condition (i) for the set of strategies ${S}_{i}$ in Theorem 3.4 is also satisfied.
Thus, from Theorem 3.4, the game $\mathrm{\Gamma}=(N,S,P,U)$ has an extended Nash equilibrium $({\overline{x}}_{1},{\overline{x}}_{2},\dots ,{\overline{x}}_{n})\in {S}_{1}\times {S}_{2}\times \cdots \times {S}_{n}$ satisfying that for every $i=1,2,3,\dots ,n$, the following order inequality holds:
We claim that $({\overline{x}}_{1},{\overline{x}}_{2},\dots ,{\overline{x}}_{n})$ is a generalized Nash equilibrium of this game $\mathrm{\Gamma}=(N,S,P,U)$. To show this claim, we need to prove that for every $i=1,2,3,\dots ,n$, the following order inequality holds:
To this end, assume, by the way of contradiction, that there is ${z}_{i}\in {S}_{i}$ such that
Since ${P}_{i}({S}_{i},{\overline{x}}_{i})$ is an order bounded Dedekind complete sublattice of ${S}_{i}$,
Therefore, there is ${y}_{i}\in {S}_{i}$ such that
From (17), the supremum in (18) implies
It is a contradiction to (16) and this corollary is proved. □
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Acknowledgements
The first author was partially supported by National Natural Science Foundation of China (11171137). The third author was partially supported by a grant from NSC 1022115M037001.
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Keywords
 poset
 lattice
 orderpreserving mapping
 fixed point
 nonmonetized noncooperative game
 generalized Nash equilibrium
 extended Nash equilibrium