Extended Nash Equilibria of Nonmonetized ... - World Scientific

2nd Reading February 25, 2015 15:20 WSPC/0219-1989 151-IGTR

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International Game Theory Review Vol. 17, No. 1 (2015) 1540009 (13 pages) c World Scientific Publishing Company DOI: 10.1142/S0219198915400095

Extended Nash Equilibria of Nonmonetized Noncooperative Games on Preordered Sets Jinlu Li∗ Department of Mathematics Shawnee State University Portsmouth, Ohio 45662, USA [email protected]

Received 13 January 2013 Revised 23 October 2013 Accepted 12 September 2014 Published 2 March 2015 A noncooperative game is said to be nonmonetized if the ranges of the utilities (payoffs) of the players are preordered sets. In this paper, we examine some nonmonetized noncooperative games in which both of the collection of strategies and the ranges of the utilities for the players are preordered sets. Then, we spread the concept of extended Nash equilibria of noncooperative games from posets to preordered sets. By applying some fixed point theorems on preordered sets and by using the order preserving property of the utilities, we prove an existence theorem of extended Nash equilibria for nonmonetized noncooperative games. Keywords: Preordered set; order preserving mapping; generalized Nash equilibrium; extended Nash equilibrium. Subject Classification: 46B42, 47H10, 58J20, 91A10

1. Introduction In traditional game theory, variational inequality theory, optimization theory and other fields, the considered functions are real valued functions or real set-valued functions (see Agarwal et al. [2012], Aubin [1979], Bohnenblust and Karlin [1950], Debreu [1959], Mas-Colell et al. [1995], Samuelson [1947], Von Neumann and Morgenstern [1944]). The fixed point theory in topological spaces or metric spaces has been an essential tool for these branches (see Carl and Heikkil¨ a [2010]). As a matter of fact, in economic theory, social science, military science, or other related fields, the outcomes of some games may not be in a linearly (totally) ordered set; that is, the utilities of the players may not be represented by real valued functions. So the ∗ Current

address: 940 Second Street, Portsmouth, Ohio 45662, USA. 1540009-1


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utility optimization problems involved with these games are different from that in most noncooperative games in game theory. In the aspect of optimization theory and variational inequality theory, Giannessi [1980] introduced the vector variational inequalities and vector optimization problems in finite-dimensional metric spaces. In 2005, Huang and Fang [2005] extended these concepts to Banach spaces. Very recently, Li [2013] introduced the concepts of nonmonetized noncooperative games and generalized Nash equilibrium on Banach lattices. Then Li and Park, and Li et al. spread the concept of generalized Nash equilibrium to more widely concepts of extended Nash equilibrium on lattices and posets, respectively, on which without any topological structure applied. In more general games, some different strategies chosen by the players may cause indifferent results; and some different outcomes of these games may be considered to have indifferent preference by the players, which are also acting as the decision makers of these games. Mathematically speaking, the strategy sets and the outcome spaces are preordered sets, which may not be posets. Based on these motivations, in this paper, we study the concepts of extended Nash equilibrium in some noncooperative games, in both of which the strategy sets and the outcome spaces are preordered sets. In Sec. 2, we recall some properties and concepts of preordered sets and posets. We also provide some connections between the poset induced by a preordered set and the given preordered set. In Sec. 3, we use some fixed point theorems in posets to prove an existence theorem of extended Nash equilibrium of games on preordered sets.

2. Preliminaries In this section, we recall and provide some concepts and properties of posets induced by some preordered sets. Here, we closely follow the notations from Agarwal et al. [2012], Aliprantis and Burkinshaw [2006], Carl and Heikkil¨ a [2010], Debreu [1959], Glicksberg [1952], Mas-Colell et al. [1995] and Ok [forthcoming]. Let P be a nonempty set. A relation on P is said to be a preorder, whenever it satisfies the following two conditions: (1) (reflexive) x x, for every x ∈ P . (2) (transitive) x y and y z imply x z, for all x, y, z ∈ P . Then P , together with the preorder , is called a preordered set, which is denoted by (P, ). Furthermore, a preorder on a preordered set P is said to be a partial order if, in addition to the above two properties, it is also satisfies (3) (antisymmetric) x y and y x imply x = y, for every x, y ∈ P. In this case, (P, ) is simply called a poset. It is worth to mention for the clarification that a preordered set (P, ) equipped with the preorder on P is defined as a partially ordered system (p.o.s.) in Dunford and Schwartz [1958]. 1540009-2


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Let (P, ) be a preordered set and let A be a subset of P. An element u of P is called an upper bound of A if x u for each x ∈ A. If u ∈ A, then u is called a greatest element (or a maximum element) of A. The collection of all greatest elements (maximum elements) of A is denoted by Max A. A lower bound of A and a smallest element (or a minimum element) of A is denoted by Min A. If the set of all upper bounds of A has a smallest element, we call it a supremum of A and the collection of all supremum elements of A is denoted by Sup A or ∨ A. An infimum of A is similarly defined as a greatest element of the set of all lower bounds of A, provided that it exists, and the collection of all infimum elements of A is denoted by Inf A or ∧ A. It is worth to note that Max A, Min A, Sup A and Inf A are subsets of P. An element y ∈ A is said to be an (efficient) maximal element of a subset A of a preordered set (P, ) if for any z ∈ A with y z implies z y. Similar to the definition of (efficient) maximal element, an (efficient) minimal element of A can be defined. A subset C of a preordered set (P, ) is said to be totally ordered (or a chain in P ) whenever for every pair x, y ∈ C either x y or y x. In particular, a totally ordered subset of a poset is just called a chain. A poset (P, ) is called a lattice if Inf{x, y} and Sup{x, y} are nonempty for all x, y ∈ P. Definition 1. An nonempty set A of a preordered set (P, ) is said to be (1) inductive if every totally ordered subset of (chain in) A has an upper bound in A; (2) inversely inductive if every totally ordered subset of (chain in) A has a lower bound in A; (3) bi-inductive whenever it is both of inductive and inversely inductive. Definition 2. An nonempty set A of a preordered set (P, ) is said to be (1) chain upper complete or strongly inductive whenever, for every totally ordered subset C of A, the set Sup C is a nonempty subset of A; (2) chain lower complete or strongly inversely inductive whenever, for every totally ordered subset C of A, the set Inf C is a nonempty subset of A; (3) chain complete or strongly bi-inductive whenever it is both of strongly inductive and strongly inversely inductive. Some authors define upper chain complete (strongly inductive) posets to be chain complete posets. Extension of Zorn’s Lemma (Extension of Theorem I.2.7 in Dunford and Schwartz [1958]). Every bi-inductive preordered set has both of efficient maximal and efficient minimal elements. Let (P, ) be a preordered set. For any x, y ∈ P, we say that x, y are -order equivalent (-order indifferent), which is denoted by x ∼ y, whenever both x y 1540009-3


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and y x hold. It is clear that ∼ is an equivalence relation on P. Let P/ ∼, or simply P˜ , denote the collection of the equivalence classes of P with respect to the equivalence relation ∼ on P. For any x ∈ P, let [x] denote the equivalence class containing x. So x ∈ [x] ∈ P/ ∼ . It is clear that a preordered set (P, ) is a poset, if and only if [x] is a singleton, for every x ∈ P. Then an ordering relation on P˜ is defined as: [x] [y] if and only if x y, for every [x], [y] ∈ P˜ .

(1)

We can see that the ordering relation defined in (1) is a partial order on P˜ ; and therefore (P˜ , ) is a poset. In general, there are some useful connections between a preordered set (P, ) and the poset (P˜ , ). We list them below as results without proof. Lemma 1. For every preordered set P = (P, ), (P˜ , ) is a poset equipped with the partial order on P˜ defined by (1), which is called the poset induced by (P, ). Furthermore, for any subset {xβ : β ∈ Λ} of P, with the index set Λ, the following properties hold : (1) z is an upper bound of {xβ : β ∈ Λ} in P if and only if [z] an upper bound of {xβ : β ∈ Λ} in P˜ ; z is a lower bound of {xβ : β ∈ Λ} in P if and only if [z] is a lower bound of {xβ : β ∈ Λ} in P˜ . (2) z ∈ Max{xβ : β ∈ Λ} in P if and only if [z] = Max{[xβ ] : β ∈ Λ} in P˜ , and z ∈ Min{xβ : β ∈ Λ} in P if and only if [z] = Min{xβ : β ∈ Λ} in P˜ . (3) z ∈ ∨{xβ : β ∈ Λ} in P if and only if [z] = ∨{[x]β : β ∈ Λ} in P˜ , and z ∈ ∧{xβ : β ∈ Λ} in P if and only if [z] = ∧{[xβ ] : β ∈ Λ} in P˜ . (4) {xβ : β ∈ Λ} is a totally ordered subset of P, if and only if {[xβ ] : β ∈ Λ} is a chain in P˜ . Given two preordered sets (X, X ) and (U, U ), we say that a single valued mapping F : X → U is order-increasing if x X y in X implies F (x) U F (y) in U. An order-increasing single valued mapping F : X → U is said to be strictly order-increasing whenever x ≺X y implies F (x) ≺U F (y). Order-increasing mappings from a preordered set to a preordered set have the following equivalence classes preserving property. Lemma 2. Let (X, X ) and (U, U ) be two preordered sets and let F : X → U be an order-increasing single valued mapping. Then, for every x in X, y ∈ [x] implies F (y) ∈ [F (x)] in U. Proof. Without any confusion, [x] is understood to be an equivalence class in (X, X ), with respect to the preorder X and [F (x)] is understood to be an equivalence class in (U, U ), with respect to the preorder U . For every x in X, y ∈ [x] if and only if both of x X y and y X x hold in X. Then the order-increasing 1540009-4


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property of F implies that both F (x) U F (y) and F (y) U F (x) hold in U. That is, F (y) ∈ [F (x)]. The proof of the following lemma is straightforward and is omitted here. ˜ X ) and Lemma 3. Let (X, X ) and (U, U ) be two preordered sets and let (X, U X U ˜ (U , ) be the posets induced by (X, ) and (U, ), respectively. For any given ˜ →U ˜ order-increasing single valued mapping F : X → U, we define a mapping F˜ : X as follows: F˜ ([x]) = [F (x)],

˜ for all [x] ∈ X.

(2)

˜ X ) to the Then F˜ is an order-increasing single valued mapping from the poset (X, ˜ , U ). poset (U It is important to note that if the mapping F : X → U is not onto, then the following strictly including property may hold: {F (y) : y ∈ [x]} ∈ [F (x)] = F˜ ([x]),

˜ for every [x] ∈ X.

(3)

In Sec. 3, we use Theorem 2.12 in Carl and Heikkil¨ a [2010] to prove the existence theorem of extended Nash equilibrium of nonmonetized noncooperative games on preordered sets. For the convenience, we recall some concepts below related to this theorem. For more details, the readers are referred to Carl and Heikkil¨ a [2010]. Let (P, ) be a preordered set. For any z, w ∈ P, we denote the -order intervals in P as follows: [z) = {x ∈ P : x z}, (w] = {x ∈ P : x w},

and

[z, w] = [z) ∪ (w] = {x ∈ P : z x w}. Given preordered sets (P, P ) and (Q, Q ), we say that a set-valued mapping F : P → 2Q \ ∅ is order-increasing upward if x X y in P and z ∈ F (x) imply that [z) ∪ F (y) is nonempty, that is, if x P y in P and z ∈ F (x), then there is w ∈ F (y) such that z Q w. F is order-increasing downward if x P y in P and w ∈ F (y) imply that (w] ∪ F (x) is nonempty, that is, if x P y in P and w ∈ F (y), then there is z ∈ F (x) such that z Q w. A set-valued mapping F is said to be order-increasing whenever F is both of order-increasing upward and downward. A nonempty subset A of a subset Y of a preordered set (P, ) is said to be order compact upward in Y if for every chain C of Y that has a supremum in P, then the intersection ∩{[y) ∩ A : y ∈ C} is nonempty whenever [y) ∩ A is nonempty for every y ∈ C. A is order compact downward in Y if for every chain C of Y that has the infimum in P the intersection ∩{[y) ∩ A : y ∈ C} is nonempty whenever [y) ∩ A is nonempty, for every y ∈ C. If A is both of 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 preordered set (P, ). An element c ∈ P is said to be a sup-center of A in P if c ∧ x exists in P for each x ∈ A. If c ∨ x exists in P for each 1540009-5


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x ∈ A, then c is said to be an inf-center of A in P. If c is both of a sup-center and an inf-center 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 sup-center or an inf-center of P, respectively. Let A be a nonempty subset of a preordered set (P, ). 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. It is clear that every nonempty strongly bi-inductive subset of a preordered set (P, ) is order closed. Now we recall the fixed point theorem on posets from Carl and Heikkil¨ a [2010]. Theorem 2.12 (Carl and Heikkil¨ a [2010]). Let (P, ) be a poset. Assume that a set-valued mapping F : P → 2P \∅ 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 sup-center or an inf-center in P, then F has (efficient ) minimal and maximal fixed points.

3. Nonmonetized Noncooperative Games on Preordered Sets In this section, we spread the concepts of nonmonetized noncooperative games and extended Nash equilibrium from posets to preordered sets. Notice that Theorem 2.12 in Carl and Heikkil¨ a [2010] is applied to posets. So in order to apply this theorem to nonmonetized noncooperative games on preordered sets, we use definitions (1) and (2) to convert these games to games on posets. Then, we apply Theorem 2.12 in Carl and Heikkil¨ a [2010] and Theorem 3.4 in Li et al. to prove an existence theorem of extended Nash equilibrium for nonmonetized noncooperative games on preordered sets. Definition 3. Let n be a positive integer greater than 1. An n-person nonmonetized noncooperative game is consisting of the following elements: (1) the set of n players, which is denoted by N = {1, 2, 3, . . . , n}; (2) the collection of n strategy sets S = {S1 , S2 , . . . , Sn }, for the n players respectively, which is also written as S = S1 × S2 × · · · × Sn ; (3) the outcome space (U, U ) that is a preordered set; (4) the n utilities functions (payoff mappings) P1 , P2 , . . . , Pn , where Pi is the utility function for player i that is a mapping from S = S1 × S2 × · · · × Sn , for every i. We write P = {P1 , P2 , . . . , Pn }. This game is denoted by Γ = (N, S, P, U ). In a n-person nonmonetized noncooperative game Γ = (N, S, P, U ), when all the n players simultaneously and independently choose their own strategies x1 , x2 , . . . , xn , respectively, to play, where xi ∈ Si , then player i will receive his utility (payoff) Pi (x1 , x2 , . . . , xn ) ∈ U, for i = 1, 2, 3, . . . , n. For any x = 1540009-6


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(x1 , x2 , . . . , xn ) ∈ S, and for every given i = 1, 2, 3, . . . , n, as usual, we denote x−i := (x1 , x2 , . . . , xi−1 , xi+1 , . . . , xn )

and

S−i := S1 × S2 × · · · × Si−1 × Si+1 · · · × Sn . Then x−i ∈ S−i and x can be simply written as x = (xi , x−i ). Moreover, we denote Pi (Si , x−i ) = {Pi (ti , x−i ) : ti ∈ Si }. Definition 4. In an n-person nonmonetized noncooperative game Γ = (N, S, P, U ), ˆ2 , . . . , x ˆn ) ∈ S = S1 × S2 × · · · × Sn , is called a selection of strategies (ˆ x1 , x (1) a generalized Nash equilibrium of this game if, for every i = 1, 2, 3, . . . , n, the following order inequality holds Pi (xi , xˆ−i ) U Pi (ˆ xi , x ˆ−i ),

for all xi ∈ Si ;

(2) an extended Nash equilibrium of this game if, for every i = 1, 2, 3, . . . , n, the following order inequality holds Pi (xi , xˆ−i ) U Pi (ˆ xi , x ˆ−i ),

for all xi ∈ Si .

Here, as usual, for a given partial order , we write x y if x y holds and y x does not hold. We write x y if y x does not hold. We need some properties of product of preordered sets and the induced orders for the proof of the main theorem in this section, which are stated below as lemmas. The proofs of these lemmas are straightforward; and therefore they are omitted here. Lemma 4. Let (Si , i ) be a preordered set, for every i = 1, 2, . . . , n. Let S = S1 × S2 × · · · × Sn be the Cartesian product space of S1 , S2 , . . . , Sn . Let S be the coordinate preorder on S induced by the preorders i , that is, for any x, y ∈ S with x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ), x S y if and only if xi i yi ,

for all i = 1, 2, . . . , n.

Then (S, S ) is a preordered set. Furthermore, if every (Si , i ) is (strongly) inductive, then (S, S ) is also (strongly) inductive. If every (Si , i ) is (strongly) bi-inductive, then (S, S ) is also (strongly) bi-inductive. Lemma 5. Let (Si , i ) be a preordered set, for every i = 1, 2, . . . , n, and let (S, S ) be the preordered set defined in Lemma 4. Then S˜ = S˜1 × S˜2 × · · · × Sñ . Moreover, for any x = (x1 , x2 , . . . , xn ) ∈ S, we have [x] = ([x1 ], [x2 ], . . . , [xn ]), where [x] stands for an equivalence class in S˜ and [x1 ], [x2 ], . . . , [xn ] are equivalence classes in S˜1 , S˜2 , . . . , Sñ , respectively. In Li et al., an existence theorem of extended Nash equilibrium of nonmonetized noncooperative games on posets is proved. We list it below. 1540009-7


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Theorem 3.4 (Li et al.). Let Γ = (N, S, P, U ) be an n-person nonmonetized noncooperative game with U a poset. If, for every player i = 1, 2, . . . , n, his strategy set (Si , i ) satisfies: (1) (Si , i ) is an strongly inversely inductive poset, (2) (Si , i ) has a sup-center (or an inf -center ) simultaneously for all i; and his utility function Pi satisfies the following conditions: (1) Pi : S → U is (single valued ) order increasing with respect to the product order S , (2) for any fixed x−i ∈ S−i , Pi (Si , x−i ) is an inductive subset of (U, U ), (3) for any fixed x−i ∈ S−i , and for any u ∈ Pi (Si , x−i ), the inverse image {zi ∈ Si : Pi (zi , x−i ) = u} is a strongly bi-inductive subset of Si , (4) for any (xi , x−i ), (yi , y−i ) ∈ S satisfying (xi , x−i ) S (yi , y−i ), the maximal elements have the following monotone properties: (a) if Pi (zi , x−i ) is a maximal element of Pi (Si , x−i ), then there is wi ∈ Si with zi i wi such that Pi (wi , yi ) is a maximal element of Pi (Si , yi ), (b) if Pi (wi , yi ) is a maximal element of Pi (Si , yi ), then there is zi ∈ Si with zi i wi such that Pi (zi , xi ) is a maximal element of Pi (Si , xi ). Then this nonmonetized noncooperative game has an extended Nash equilibrium. Furthermore, this nonmonetized noncooperative game has minimal and maximal (with respect to the product lattice order S ) extended Nash equilibria. Now we extend the above theorem from posets to preordered sets. It is the main theorem of this paper. Theorem 1. Let Γ = (N, S, P, U ) be an n-person nonmonetized noncooperative game. If, for every player i = 1, 2, . . . , n, his strategy set Si satisfies: (1) Si , i ) is a strongly inversely inductive preordered set, (2) (Si , i ) has a sup-center (or an inf -center), simultaneously for all i, and his utility function Pi satisfies the following conditions: (1) Pi : S → U is a single-valued order-increasing mapping with respect to the product order S ; (2) for any fixed x−i ∈ S−i , Pi (Si , x−i ) is an inductive subset of (U, U ); (3) for any fixed x−i ∈ S−i , and for any u ∈ Pi (Si , x−i ), the inverse image {zi ∈ Si : Pi (zi , x−i ) = u} is a strongly bi-inductive subset of Si ; (4) for any (xi , x−i ), (yi , y−i ) ∈ S satisfying (xi , x−i ) S (yi , y−i ), the maximal elements of Pi (Si , x−i ) have the following monotonic properties: (a) if Pi (zi , x−i ) is a maximal element of Pi (Si , x−i ) then there is wi ∈ Si with zi i wi such that Pi (wi , y−i ) is a maximal element of Pi (Si , y−i ), (b) if Pi (wi , y−i ) is a maximal element of Pi (Si , y−i ), then there is zi ∈ Si with zi i wi such that Pi (zi , x−i ) is a maximal element of Pi (Si , x−i ). 1540009-8


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Then this nonmonetized noncooperative game Γ has an extended Nash equilibrium. Furthermore, this nonmonetized noncooperative game Γ has (efficient ) minimal and maximal (with respect to the product order S ) extended Nash equilibria. Proof. For every i = 1, 2, 3, . . . , n, let (S˜i , i ) denote the poset induced by (Si , i ), ˜ S ) and (U ˜ , U ) respectively denote the posets induced by respectively. Let (S, (S, S ) and (U, U ). For every payoff function Pi : S → U, i = 1, 2, 3, . . . , n, we ˜ following Eq. (2) as below: define a function P˜i : S˜ → U P˜i ([x]) = [Pi (x)],

˜ for all [x] ∈ S.

(4)

˜ and U ˜ , respectively, Now we show that if, Si , S and U are substituted by S˜i , S, ˜ , for i = 1, 2, 3, . . . , n, in Theorem 3.4 in and Pi : S → U, is replaced by P˜i : S˜ → U ˜ U ˜ and P˜i will satisfy Li et al., then the conditions of this theorem ensure that S˜i , S, all conditions in Theorem 3.4 in Li et al. Since every strategy set (Si , i ) is a preordered set, then from Lemma 1, (S˜i , i ) is a poset equipped with the partial order i on S˜i defined by (1). We show that the two conditions on Si in this theorem ensure that the collections of the equivalence classes S˜i satisfy the conditions i and ii in Theorem 3.4 in Li et al. To this end, for any given chain {[xiα ] : α ∈ Λ} in S˜i , with the index set Λ, {xiα } is a totally ordered subset of (a chain in) S˜i . From Lemma 1, we have zi is an upper bound of {xiα : α ∈ Λ} in Si if and only if [zi ] is an upper bound of {xiα : α ∈ Λ} in S˜i ; zi is a lower bound of {xiα : α ∈ Λ} in Si if and only if [zi ] is a lower bound of {xiα : α ∈ Λ} in S˜i .

(5)

It implies that Si is a strongly inversely inductive preordered set if and only if it is S˜i is a strongly inversely inductive poset. Hence the condition i in this theorem ensures that condition i in Theorem 3.4 in Li et al. holds for (S˜i , i ). In particular in Lemma 1, for any pair elements xi and yi in Si , zi ∈ xi ∨ yi in Si (or zi ∈ xi ∧ yi in Si ) if and only if [zi ] = [xi ] ∨ [yi ] in S˜i (or [zi ] = [xi ] ∧ [yi ] in S˜i ). So ai is a sup-center (or an inf -center) of (Si , i ) if and only if [ai ] is a sup-center (or an inf -center) of (S˜i , i ). Then the condition ii of this theorem implies that condition ii of Theorem 3.4 in Li et al. holds with respect to (S˜i , i ). ˜ satisfies the conditions 1 through 4 Next we show that the mapping P˜i : S˜ → U in Theorem 3.4 in Li et al. If Pi : S → U satisfies condition 1 in this theorem, then ˜ is an order-increasing mapping from (S, ˜ S ) to (U ˜ , U ), from Lemma 3, P˜i : S˜ → U ˜ for i = 1, 2, 3, . . . , n. So Pi satisfies condition 1 in Theorem 3.4 in Li et al. For any fixed [x−i ] ∈ S˜−i , which is equivalent to xi ∈ Si , from condition 2 of this theorem, 1540009-9


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Pi (Si , xi ) is an inductive subset of (U, U ). Applying (5), it yields that P˜i (S˜i , [x−i ]) ˜ , U ), which proves that P˜i satisfies condition 2 in is an inductive subset of (U Theorem 3.4 in Li et al. For any given [x−i ] ∈ S˜−i , and [u] ∈ P˜i (S˜i , [x−i ]), we have x−i ∈ S−i . From the inclusion formula (3), the condition [u] ∈ P˜i (S˜i , [x−i ]) does not imply u ∈ Pi (Si , x−i ). But the condition [u] ∈ P˜i (S˜i , [x−i ]) ensures that there is an v ∈ [u] such that v ∈ Pi (Si , x−i ), which satisfies {[zi ] ∈ S˜i : P˜i ([zi ], [x−i ]) = [u]} = {[zi ] ∈ S˜i : P˜i ([zi ], [x−i ]) = [v]}.

(6)

Since Pi : S → U is an order-increasing mapping, from Lemma 2, Pi has the equivalence classes reserving property. Then for any zi ∈ Si , we have that Pi (zi , x−i ) = v

if and only if P˜i ([zi ], [x−i ]) = [v].

(7)

Condition 3 in this theorem states that {zi ∈ Si : Pi (zi , x−i ) = v} is a strongly bi-inductive subset of Si . From (5) and (6), it implies that {[zi ] ∈ S˜i : P˜i ([zi ], [x−i ]) = [v]} is a strongly bi-inductive subset of S˜i . Then, from Eq. (6), {[zi ] ∈ S˜i : P˜i ([zi ], [x−i ]) = [u]} is a strongly bi-inductive subset of S˜i . So P˜i satisfies condition 3 in Theorem 3.4 in Li et al. ˜ which are equivalent to (xi , x−i ), (yi , y−i ) ∈ For any ([xi ], [x−i ]), ([yi ], [y−i ]) ∈ S, S S, ([xi ], [x−i ]) ([yi ], [y−i ]), implies (xi , x−i ) S (yi , y−i ). If P˜i ([zi ], [x−i ]) is a maximal element of P˜i (S˜i , [x−i ]), from (5), then Pi (zi , x−i ) is a maximal element of Pi (Si , x−i ). From part (a) of condition 4 in this theorem, there is wi ∈ Si with zi i wi such that Pi (wi , y−i ) is a maximal element of Pi (Si , y−i ). Applying (5) again, P˜i ([wi ], [y−i ]) is a maximal element of P˜i (S˜i , [yi ]). The condition zi i wi is equivalent to [zi ] i [wi ]. Hence part (a) of condition 4 in Theorem 3.4 in Li et al. is satisfied with respect to P˜i and S˜i . Similar to the proof of part (a), we can show that part (b) of condition 4 in Theorem 3.4 in Li et al. is also satisfied with respect to P˜i and S˜i . ˜ U ˜ and the utility functions P˜i : S˜ → Hence we proved that the posets S˜i , S, ˜ , i = 1, 2, 3, . . . , n, satisfy all conditions in Theorem 3.4 in Li et al. Thus we obtain U that there is a selection of equivalence classes of strategies ([ˆ x1 ], [ˆ x2 ], . . . , [ˆ xn ]) ∈ S˜ = S˜1 × S˜2 × · · · × Sñ such that, for every i = 1, 2, 3, . . . , n, the following orderinequality holds P˜i ([xi ], [ˆ x−i ]) U P˜i ([ˆ xi ], [ˆ x−i ]),

for all [xi ] ∈ S˜i .

Applying the order-increasing property of Pi , and using Lemmas 3 and 5, it is equivalent to xi , x ˆ−i ), P˜i (xi , xˆ−i ) U P˜i (ˆ

for all xi ∈ Si .

x2 ], . . . , [ˆ xn ]) ∈ S˜ = S˜1 × S˜2 × · · · × Sñ such Thus the selection of strategies ([ˆ x1 ], [ˆ that, for every i = 1, 2, 3, . . . , n, is an extended Nash equilibrium of this game Γ = (N, S, P, U ). 1540009-10


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4. Conclusion In game theory, for any n-person noncooperative strategy game, if all payoff functions take values of real numbers, which is a lattice with the ordinary real order, then it is a special case of n-person nonmonetized noncooperative games defined in Definition 3. In order to show the concrete practice of the concepts of n-person nonmonetized noncooperative games, generalized Nash equilibria, and generalized Nash equilibria, we demonstrate the following example to conclude our paper. This example is very similar to Example 5.2 in Li [2013], which extends the prisoners dilemma. Let R2 denote the 2-dimensional Euclidean space. Let 2 denote the coordinate ordering relation on R2 , which is defined as x 2 y if an only if xj ≥ yj , for j = 1, 2, x = (x1 , x2 ), y = (y1 , y2 ) ∈ R 2 . Then 2 is a partial order on R 2 and (R 2 , 2 ) is a poset. Example (the extended prisoners dilemma — complicated version). Two suspects, designated Suspect 1 and Suspect 2, are held in separate cells without any means of communicating with each other. There are two crimes (I and II) for which these suspects are being held. There is enough evidence to convict each of them of minor offenses related to crimes I and II, but not enough evidence to convict either of them of the principal crimes I or II unless one of them acts as an informant against the other (finks) for major crime I or II. If they both stay quiet for both crimes I and II, then each will be convicted of both minor offenses, and will spend one year in prison for crime I and fined $10 for crime II. If only one finks for crime I and both stay quiet for crime II, then the informant will not be charged for crime I but will be fined $10 for crime II, and the informant will testify against the other suspect, who will be convicted of the principal offense for crime I resulting in a three-year prison sentence and also be fined $20 for crime II. If they both stay quiet for crime I and only one of them finks for crime II, then the informant will not be charged for crime II but will spend one year in prison for crime I, and the informant will testify against the other suspect, who will be convicted of the principal offense for crime II resulting in a $30 fine and also sentenced to two years in prison for crime I. If one and only one of them finks for both crimes I and II, and other one stays quiet for both crime I and crime II, then the informant will be charged for neither crime I nor crime II, and will testify against the other for both crimes, who will be convicted of the offense for crime I and will spend three years in prison for the crime I and will be fined $30 for crime II. If both suspects fink for both crimes I and II, then each will spend two years in prison for crime I and be fined $20 for crime II. Every suspect has four possible strategies to apply: QQ, QF , FQ, FF , where the first letter stands for the action on crime I and the second one is for the action on crime II. Then the possible outcomes (payoffs) for this game can be described by the following table, where Suspect 1 plays this game as the row player and Suspect 2 plays this game as the column player. 1540009-11


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J. Li Table 1.

The utilities of the two players in the extended prisoners dilemma.

QQ QQ (−1, −10), (−1, −10) QF (−1, 0), (−2, −30) FQ FF

(0, −10), (−3, −20) (0, 0), (−3, −30)

QF

FQ

FF

(−2, −30), (−1, 0) (−2, −20), (−2, −20)

(−3, −20), (0, −10) (−3, −10), (−1, −30)

(−3, −30), (0, 0) (−3, −20), (−1, −20)

(−1, −30), (−3, −10) (−1, −20), (−3, −20)

(−2, −20), (−2, −20) (−2, −10), (−3, −30)

(−3, −30), (−2, −10) (−2, −20), (−2, −20)

In every entry of the above matrix table, the first point is the outcome for Suspect 1 and the second point is for Suspect 2. At each point, the absolute value of the first coordinate defines the number of years for the suspect to spend in prison for convicted offense I and the absolute value of the second coordinate shows the amount of dollars for the suspect to be fined for convicted offense II. Let u1 , u2 be the utility functions (payoffs) of Suspect 1 and Suspect 2, respectively. Suppose that the number of years in prison for convicted offense I and the number of dollars imposed for the fine of convicted offense II are not substitutable. So the preferences of the possible outcomes for every suspect are not totally ordered, and both u1 , and u2 take values in the poset (R 2 , 2). Then we have u1 (QQ, QQ) = (−1, −10), u1 (FQ, QQ) = (0, −10),

u1 (QQ, QF ) = (−2, −30), u1 (FF , QQ) = (0, 0),

and

u1 (QF , FQ ) = (−3, −10). It implies u1 (QQ, QF ) ≺2 u1 (QQ, QQ) ≺2 u1 (FQ, QQ) ≺2 u1 (FF , QQ) and u1 (QQ, QF ) ≺2 u1 (QF , FQ). It can be easily checked that the action (strategy) profile (FF , FF ) is a generalized Nash equilibrium of this game. Acknowledgment The author is grateful for the anonymous reviewers for their valuable suggestions, which really improved the presentation of this paper. References Agarwal, R. P., Balej, M. and O´Regan, D. [2012] A unifying approach to varational relation problems, J. Optimal Theory Appl. 154, 229–231. Aliprantis, C. D. and Burkinshaw, O. [2006] Positive Operators (Springer, The Netherland). Aubin, J. P. [1979] Mathematical Methods of Games and Economic Theory (Nover Publications, North-Holland). 1540009-12


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Bohnenblust, H. F. and Karlin, S. [1950] Contributions to the Theory of Games (Princeton University Press, NJ, USA). Carl, S. and Heikkil¨ a, S. [2010] Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory (Springer, New York). Debreu, G. [1959] Theory of Value: An Axiomatic Analysis of Economic Equilibrium (Yale University Press, New Heven). Dunford, N. and Schwartz, J. T. [1958] Linear Operators, Part I : Genearal Theory (John Wiley and Sons, NJ, USA). Huang, N. J. and Fang, Y. P. [2005] On vector variational inequalities in reflexive banach spaces, J. Global Optim. 32, 495–505. Giannessi, F. [1980] Theorems of alterative, quadratic progams and complementarity problems, in Variational Inequalities and Complementarity Problems, Wiley, NY, pp. 151–186. Glicksberg, I. [1952] A further generalizedization of the Kakutani fixed point theorem with applications to Nash equilibrium points, Proc. Amer. Math. Soc. 3, 170–174. Li, J. L. [2013] Applications of fixed point theory to generalized Nash-equilibriums of nonmonetized noncooperative games on banach lattices, J. Nonlinear Anal. Forum 18, 1–11. Li, J. L. and Park, S. [2014] Generalized Nash-equilibriums of non-monetized noncooperative games on lattices, Br. J. Econ. Manag. Trade 4, 97–110. Li, J. L., Xie, L. S. and Wen, Y. C., Extended Nash-equilibriums of nonmonetized noncooperative games on posets, to appear in Fixed Point Theory Appl. 2013, 2013/1/235. Mas-Colell, A., Whinston, M. D. and Green, J. R. [1995] On Vector Variational Inequalities in Reflexive Banach Spaces, Microeconomic Theory (Oxford University Press, Oxford, UK). Ok, E. A., Order theory, forthcoming. Samuelson, P. [1947] Foundations of Economic Analysis (Harvard University Press, Cambridge, MA, USA). Von Neumann, J. and Morgenstern, O. [1944] The Theory of Games and Economic Behavior (Princeton University Press, NJ, USA).

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