Fuzzy Game Theory: a Novel Perspective Vahid Behravesh*, S.M.R

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Volume 2, Issue 7, July 2012 ISSN: 2277 128X International Journal of Advanced Research in Computer Science and Software Engineering Research Paper Available online at: www.ijarcsse.com Fuzzy Game Theory: A Novel Perspective Vahid Behravesh*, S.M.R. Farshchi Islamic Azad University,Bardaskan Branch, Department of Electrical Engineering, Bardaskan, Iran

Abstract— Involving of uncertainty into cooperative games is motivated by the real world where noise in observation and experimental design, incomplete information and further vagueness in preference structures and decision making play an important role. The theory of cooperative ellipsoidal games provides a new game theoretical understanding and suitable tools for to solve this question. This research aims to brieﬂy present the state-of-the-art in this young ﬁeld of research, discusses how the model of cooperative ellipsoidal games extends the cooperative game theory literature, and reviews its application in real life.

Keywords— Game Theory, Fuzzy Game, Evolutionary Game.

I. INTRODUCTION Combinatorial game theory (CGT) is a branch of applied optimal solution existence without necessarily specifying an mathematics and theoretical computer science that studies algorithm (see strategy-stealing argument for instance). sequential games with perfect information, that is, two-player An important notion in CGT is that of solved game (which games which have a position in which the players take turns has several flavors), meaning for example that one can prove changing in defined ways or moves to achieve a defined that the game of tic-tac-toe results in a draw if both players winning condition. CGT does not study games with imperfect play optimally. While this is a trivial result, deriving similar or incomplete information (sometimes called games of chance, results for games with rich combinatorial structures is difficult. like poker). It restricts itself to games whose position is public For instance, in 2007 it was announced that checkers has to both players, and in which the set of available moves is also been (weakly, but not strongly) solved—optimal play by both public (perfect information). Combinatorial games include sides also leads to a draw—but this result was a computer- well-known games like chess, checkers, Go, Arimaa, Hex, and assisted proof. Other real world games are mostly too Connect6. They also include one-player combinatorial puzzles, complicated to allow complete analysis today (although the and even no-player automata, like Conway's Game of Life. [2] theory has had some recent successes in analyzing Go In CGT, the moves in these games are represented as a endgames). Applying CGT to a position attempts to determine game tree [1-2]. the optimum sequence of moves for both players until the Game theory in general includes games of chance, games game ends, and by doing so discover the optimum move in of imperfect knowledge, and games in which players can any position. In practice, this process is tortuously difficult move simultaneously, and they tend to represent real-life unless the game is very simple. decision making situations. CGT has a different emphasis than "traditional" or "economic" game theory, which was initially II. DEFINITION developed to study games with simple combinatorial structure, A game, in its simplest terms, is a list of possible "moves" but with elements of chance (although it also considers that two players, called left and right, can make. The game sequential moves, see extensive-form game) [3]. Essentially, position resulting from any move can be considered to be CGT contributed new methods for analyzing game trees, for another game. This idea of viewing games in terms of their example using surreal numbers, which are a subclass of all possible moves to other games leads to a recursive two-player perfect-information games.[2] mathematical definition of games that is standard in The type of games studied by CGT is also of interest [4] combinatorial game theory. In this definition, each game has in artificial intelligence, particularly for automated planning the notation L|R = {R} is the set of game positions that the left and scheduling. In CGT there has been less emphasis on player can move to, and R is the set of game positions that the refining practical search algorithms (like the alpha-beta right player can move to; each position in L and R is defined pruning heuristic, included in most artificial intelligence as a game using the same notation. textbooks today), but more emphasis on descriptive theoretical Use Domineering as an example, label each of the sixteen results, like measures of game complexity or proofs of boxes of the four-by-four board by A1 for the upper leftmost

Volume 2, Issue 7, July 2012 www.ijarcsse.com square, C2 for the third box from the left on the second row The first player (Left or Right) wins. from the top, and so on. We use e.g. (D3, D4) to stand for the game position in which a vertical domino has been placed in Using standard Dedekind-section game notation, {L|R}, the bottom right corner. Then, the initial position can be where L is the list of undominated moves for Left and R is the described in combinatorial game theory notation as list of undominated moves for Right, a fuzzy game is a game where all moves in L are strictly non-negative, and all moves in R are strictly non-positive. One example is the fuzzy game * = {0|0}, which is a first- player win, since whoever moves first can move to a second Note that, in standard Cross-Cram play, the player’s player win, namely the zero game. An example of a fuzzy alternate turns, but this alternation is handled implicitly by the game would be a normal game of Nim where only one heap definitions of combinatorial game theory rather than being remained where that heap includes more than one object. encoded within the game states. Another example is the fuzzy game {1|-1}. Left could move to 1, which is a win for Left, while Right could move to -1, which is a win for Right; again this is a first-player win. In Blue-Red-Green Hackenbush, if there is only a green The above game (an irrelevant open square at C3 has been edge touching the ground, it is a fuzzy game because the first omitted from the diagram) describes a scenario in which there player may take it and win (everything else disappears). is only one move left for either player, and if either player In game theory, normal form is a description of a game. makes that move, that player wins. The {|} in each player's Unlike extensive form, normal-form representations are not move list (corresponding to the single leftover square after the graphical per se, but rather represent the game by way of a move) is called the zero game, and can actually be abbreviated matrix. While this approach can be of greater use in 0. In the zero game, neither player has any valid moves; thus, identifying strictly dominated strategies and Nash equilibrium, the player whose turn it is when the zero game comes up some information is lost as compared to extensive-form automatically loses. representations. The normal-form representation of a game The type of game in the diagram above also has a simple Includes all perceptible and conceivable strategies, and their name; it is called the star game, which can also be abbreviated corresponding payoffs, of each player. In the star game, the only valid move leads to the zero game, In static games of complete, perfect information, a normal- which means that whoever’s turn comes up during the star form representation of a game is a specification of players' game automatically wins. strategy spaces and payoff functions. A strategy space for a An additional type of game, not found in Domineering, is a player is the set of all strategies available to that player, where loopy game, in which a valid move of either left or right is a a strategy is a complete plan of action for every stage of the game which can then lead back to the first game. Checkers, game, regardless of whether that stage actually arises in play. for example, becomes loopy when one of the pieces promotes, A payoff function for a player is a mapping from the cross- as then it can cycle endlessly between two or more squares. A product of players' strategy spaces to that player's set of game that does not possess such moves is called loopfree. payoffs (normally the set of real numbers, where the number In combinatorial game theory, a fuzzy game is a game represents a cardinal or ordinal utility—often cardinal in the which is incomparable with the zero game: it is not greater normal-form representation) of a player, i.e. the payoff than 0, which would be a win for Left; nor less than 0 which function of a player takes as its input a strategy profile (that is would be a win for Right; nor equal to 0 which would be a a specification of strategies for every player) and yields a win for the second player to move. It is therefore a first-player representation of payoff as its output. win. In combinatorial game theory, there are four types of game. III. DOMINANCE STRATEGY If we denote players as Left and Right, and G be a game with The payoff matrix facilitates elimination of dominated some value, we have the following types of game: strategies, and it is usually used to illustrate this concept. For example, in the prisoner's dilemma (to the right), we can see 1. Left win: G > 0 that each prisoner can either "cooperate" or "defect". If No matter which player goes first, Left wins. exactly one prisoner defects, he gets off easily and the other prisoner is locked up for good. However, if they both defect, 2. Right win: G < 0 they will both be locked up for longer. One can determine that No matter which player goes first, Right wins. Cooperate is strictly dominated by Defect. One must compare the first numbers in each column, in this case 0 > −1 3. Second player win: G = 0 and −2 > −5. This shows that no matter what the column The first player (Left or Right) has no moves, and thus player chooses, the row player does better by choosing Defect. loses. Similarly, one compares the second payoff in each row; again 0 > −1 and −2 > −5. This shows that no matter what row does; 4. First player win: G ║ 0 (G is fuzzy with 0) column does better by choosing Defect. This demonstrates the

Volume 2, Issue 7, July 2012 www.ijarcsse.com unique Nash equilibrium of this game is (Defect, Defect). (Fig.  Each node of the Chance player has a probability 1) distribution over its outgoing edges. An extensive-form game is a specification of a game in  Each set of nodes of a rational player is further game theory, allowing (as the name suggests) explicit partitioned in information sets, which make certain choices representation of a number of important aspects, like the indistinguishable for the player when making a move, in the sequencing of players' possible moves, their choices at every sense that: decision point, the (possibly imperfect) information each player has about the other player's moves when he makes a o there is a one-to-one correspondence between outgoing decision, and his payoffs for all possible game outcomes. edges of any two nodes of the same information set—thus the Extensive-form games also allow representation of set of all outgoing edges of an information set is partitioned in incomplete information in the form of chance events encoded equivalence classes, each class representing a possible choice as "moves by nature". for a player's move at some point—, and

o every (directed) path in the tree from the root to a terminal node can cross each information set at most once the complete description of the game specified by the above parameters is common knowledge among the players

IV. CONCLUSION & DISCUSSION Van Huyck, et al. (1995) discovered an evolutionary bargaining game in which unequal division conventions emerge amongst symmetrically endowed subjects. They call it DS:

Fig. 1 Both extensive and normal form illustration of a sequential form game with subgame imperfect and perfect Nash equilibriium marked with red and blue respectively.

Some authors, particularly in introductory textbooks, initially define the extensive-form game as being just a game tree with payoffs (no imperfect or incomplete information), and add the other elements in subsequent chapters as refinements. Whereas the rest of this article follows this gentle approach with motivating examples, we present upfront the DS is symmetric. Units denote dimes. If both players finite extensive-form games as (ultimately) constructed here. choose 2 they divide a dollar equally. If one player chooses This general definition was introduced by Harold W. Kuhn action 1 and the other chooses action 3, the first earns $0.60 in 1953, which extended an earlier definition of von Neumann and the other earns $0.40. Game DS is unusual in that the from 1928. Following the presentation from Hart (1992), an n- aggressive demand is also the secure demand. Using action 1 Player extensive-form game thus consists of the following: insures earning $0.30. While the stage game is symmetric, Van Huyck, et al.  A finite set of n (rational) players (1995) used a two labeled population protocol, which as we have just seen, allows some cohorts to use labels and  A rooted tree, called the game tree populations to break the symmetry of the stage game in the evolutionary game.  Each terminal (leaf) node of the game tree has an n- Figure 1 graphs the logistic response vector field for the tuple of payoffs, meaning there is one payoff for each player game that results when action 2 becomes extinct in game DS, at the end of every possible play which will be denoted BOS. The black dots denote Nash equilibria and the red dots denote logistic equilibria (again  A partition of the non-terminal nodes of the game tree with equal 1). The grey shaded regions indicate states in in n+1 subsets, one for each (rational) player, and with a which everyone has a pecuniary incentive to conform to the special subset for a fictitious player called Chance (or Nature). emerging convention. It is not obvious from the figure or from Each player's subset of nodes is referred to as the "nodes of logistic response theory why action 2 goes extinct, but one the player". (A game of complete information thus has an expects it to depend on the initial condition. Once action 2 is empty set of Chance nodes.) extinct, however, the vector field implies that one of the two

Volume 2, Issue 7, July 2012 www.ijarcsse.com unequal division conventions emerges. Van Huyck, et al. [5] Daniel Friedman, ―Equilibrium in evolutionary games: discovered that the equal division action always goes extinct. Some experimental results,‖ Economic Journal, 106:1-25, 1996.

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[4] Russell Cooper, Douglas V. DeJong, Robert Forsythe, and Thomas W. Ross, ―Communication in coordination games,‖ Quarterly Journal of Economics, 107:739-773, 2010.