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Game Theory

Game Theory. Part 1: Introduction and Types of Games. What is Game Theory?. Game theory is the study of the strategic interaction among rational players.

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Game Theory

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  1. Game Theory Part 1: Introduction and Types of Games

  2. What is Game Theory? • Game theory is the study of the strategic interaction among rational players. • The games studied can be quite serious and are studied in many areas of the natural and social sciences: military, political and marketing campaign strategy can be modeled with game theory, also phenomena in natural science, for example, in the study of evolutionary biology. • Participants in a game are called players, sometimes one of the players could be nature or chance. Players may have one or more possible options for play. These options are referred to as moves or strategies. We study the rational selection of strategies, the interaction of players and the resulting outcomes. • We assume players are rational in the sense that they seek the outcome where the resulting payoff is in their best interest.

  3. Types of Games • Alternate-move games (like chess and tic-tac-toe) – players take turns, strategies involve many moves and can evolve during play. • Simultaneous-move games (like rock-paper-scissors) – players commit to a move without knowledge of the other players’ strategy. • Economic, political or military decisions, and many other real-life situations, can be modeled by simultaneous-move games even if the players don’t make their move at the same time. For example, once a strategy is chosen, regardless of an opponent’s move, it may be too late for a player to change strategy. One player making a move after the other may be equivalent to a simultaneous-move game if each had predetermined strategies (with which they are committed to play) chosen without knowing the intentions of the other. • In this chapter, we study simultaneous move games. However, we will consider situations where simultaneous move games are repeated and then can become similar to alternate move games.

  4. Types of Games • Zero-sum games: Games of “total conflict” – one player’s gain equals the other player’s loss. If we sum the payoffs at each outcome, the result is always zero. • Constant-sumgames: If we sum the payoffs at each outcome, the result is always the same constant. Any constant-sum game can be easily converted to a zero-sum game by an appropriate translation of the payoff amounts. Thus constant-sum games are equivalent to zero-sum games. • Variable-sum games: Games of “partial conflict” – outcomes may involve gains for both players, a gain for only one or the other, or a loss for both players. • This chapter is only a brief introduction to game theory: we study only the basics with simultaneous-move zero sum and variable sum games, there is much more to game theory than that.

  5. Matrix Games • Two-player simultaneous move games (both zero and variable sum types) can be written in matrix form (also called strategic form) as shown below. • The strategies of one player form the rows of the matrix, while the strategies of the other player form the columns. Each entry in the matrix represents a possible outcome based on a corresponding selection of strategies.

  6. Matrix Games – A 2X2 Matrix Game Player II ( column Player ) • Each outcome consists of two values which represent payoffs to each player. • For example, if the column player chooses strategy A and the row player chooses strategy X, the outcome is represented by the values (m1, m2), where m1 is the payoff for player I and m2 is the payoff for player II. Player I ( row player )

  7. Matrix Games – A 2X2 Matrix Game Player II ( Column Player ) • For all matrix games, when outcomes are written using two coordinates, let’s assume the payoff to the row player is the first coordinate, while the second coordinate represents the payoff to the column player. Player I ( row player )

  8. An Example of a 2x2 Matrix Game Player II ( Column Player ) • Consider the above example of a matrix game. • We’ll use the convention that larger payoff values are better. • Assuming player I and II move simultaneously, what should they pick? • If you are player I, what is your strategy to optimize your payoff? Do you pick X or Y? • If you are player II, what is your strategy? Is it better to pick A or B? Player I ( row player )

  9. An Example 2x2 Matrix Game Player II ( Column Player ) • Suppose you are player I and have to decide between choices X or Y (called pure strategies). • If you pick X, then you get either 1 point if player II picks A or 4 points if player II picks B. • But if you pick Y, then you get either 2 points if player II picks A or 3 points if player II picks B. Player I ( row player )

  10. A Matrix Game Player II ( Column Player ) • On the other hand, if you are player II, you are choosing between pure strategies A or B. • If you pick A, then you get a payoff of 2 points if player I picks X and a payoff of 1 if player I picks Y. • If you pick B, then you get a payoff of –3 (a loss) if player I picks X and a payoff of 3 points if player I picks strategy Y. Player I ( row player )

  11. Matrix Games – Payoffs are known by both Players Player II ( Column Player ) • Because this is a simultaneous move game, neither player knows the strategy choice of the other player until the game is played. • In matrix games, we assume both players know the payoffs associated with each strategy (and we assume both players know that both players know the payoffs associated with each strategy, etc.) Player I ( row player )

  12. Choosing a Strategy – Cyclic Reasoning Player II ( Column Player ) • In deciding which strategy to follow, player I might go through the following reasoning: • If I choose option X, then I could get the highest payoff of 4 if player II chose B. However, player II is unlikely to pick B because he could suffer a loss of 3 if I pick X. Thus player II may choose A. If player II is going to choose A then I should choose Y which is a better payoff for me when player II is playing A. But then if player II knows that I am reasoning in this way, and am therefore more likely to pick strategy Y, he may choose to pick strategy B, in which case I am better off with strategy X … Player I ( row player )

  13. Matrix Games Player II ( Column Player ) • Can we find an optimal strategy for either or both players? • The answer is yes, but before doing so, we’ll need to understand certain terms. • We’ll continue by defining some of those terms and also considering some easier special cases … Player I ( row player )

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