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15 THEORY OF GAMES CHAPTER

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Learning Objectives Assumptions, Definitions and Classification of Games. Two-person Zero Sum Games Saddle Point and Pure Strategies Dominance Mixed Strategies Graphical Solution of Games

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Theory of Games It deals with decision making under conditions of competition. It deals with situations where different parties have opposing interests and both take decisions so that there interests are promoted.

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Assumptions Conflict must be focussed on a central issue. Players (parties to the conflict) must have opposing interests or the same objective. Players act rationally and intelligently. Each player has a finite set of alternatives to choose from. Players must make simultaneous and concurrent decisions as to which of the set of alternatives to employ. These decisions, once made, cannot be reversed during a particular round of conflict. Each player must measure the worth of all possible outcomes on the same scale. Each player must exert some influence to bear on the situation but must not be able to exert complete control over the situation by himself.

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Definitions Person. The term refers to one of the opposing interests or players. It could be an individual, a group of individuals or an organisation. Game. The set of rules that defines what can and cannot be done; the payoffs in terms of monetary values or utility values and pay off methods. Rules must be complete and no change is permitted during the game. Zero Sum Game. A game in which the total winnings equal the total losses. Play of Game. The choosing of a course of action by a player along with an exchange of resulting payoffs.

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Definitions Strategy. A players course of action that is complete and ready before commencement of the game. Optimal Strategy. A strategy which guarantees a player the best he can expect regardless of what the opponent player does. Value of the Game. Expected payoff when each player is using his/her optimal strategy. Solution. An optimal strategy for each player and the value of the game.

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**Classification of Games**

If a game has two players we refer to it as a two person game. When the players are three or more, the game is referred to as an n-person game. Solution of n-person games is still being developed If the sum of total gains and losses is zero, the game is referred to as a two person zero sum game. If the sum of total gains and losses is not zero, the game is referred to as a two person non-zero sum game.

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**Two-Person Zero Sum Games**

Consider the game: Player A Player B Strategy M Strategy N Strategy X B wins 2 points B wins 1 point Strategy Y A wins 3 points A wins 4 points As can be seen, the game is heavily in favour of A. Given a choice A will play only Strategy Y, as he will always win. B will try to minimise his loss by always playing Strategy M. This is a two-person zero sum game.

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**Two-Person Zero Sum Game**

The game is written as a matrix with payoffs always written from the row player’s point of view. Player A Player B Strategy M Strategy N Strategy X B wins 2 points B wins 1 point Strategy Y A wins 3 points A wins 4 points

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**Saddle Point and Pure Strategies Example**

Blue and Red Scooters may both indulge in no advertising, moderate advertising and heavy advertising. If Blue does no advertising – it will get 50% market share if Red also does no advertising; 40% market share if Red advertise moderately and 28% market share if Red advertises heavily. Blue use moderate advertising – it will get 70% of the market share with no advertising by Red, 50% of the market share if Red also resorts to moderate advertising, and 45% of the market share if Red does heavy advertising. Blue uses heavy advertising – in such an event it gets 75% of the market share if Red does not advertise, 55% if Red advertises moderately and 50% if Red advertises heavily.

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**Saddle Point and Pure Strategies Example**

The problem may be written as: Blue Red None Moderate Heavy 50 40 28 70 45 75 55

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**Blue should advertise heavily and Red should also advertise heavily.**

Both players will attempt to maximise their gains or minimise their losses. Blue will choose a criterion of maximin – best of the worst. Red will also attempt to minimise his losses by using a similar strategy. Since the payoffs are written from Blue’s point of view, Red will follow the criterion of minimax – minimum of the maximums as the matrix represents loss or regret for him. Blue Red None Moderate Heavy Maximin 50 40 28 70 45 75 55 Minimax Blue should advertise heavily and Red should also advertise heavily. This course will always yield a payoff of 50.

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Blue Red None Moderate Heavy Maximin 50 40 28 70 45 75 55 Minimax There is a single strategy for both players which would maximise their gains (or minimise their losses). Both players will play the same strategy every time the game is played. Each player has a pure strategy, one that he will play all the time. The payoff obtained when each player plays his pure strategy is called a saddle point, or the saddle point is the value of the game when each player has a pure strategy.

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Mixed Strategies When a pure strategy is not available for the players, they will have to use a mix of strategies. It is imperative that the strategies are mixed in such a manner that the payoffs are optimised. When using mixed strategies, the players have to maintain security of their game plan. It must be ensured that the opponent does not come to know of the strategy that will be employed. If he can get prior information, he will choose an appropriate strategy to counter the one used by the other player. The strategies will have to played randomly in the desirable proportions so that no patterns of play can be established.

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**Mixed Strategies - Example**

Player A Player B b1 B2 a1 1 a2 - 4 3 Let p be the proportion of time A plays row a1 , then (1 – p) is the proportion of time that he plays a2. He would expect the same payoff irrespective of whether B plays b1 or b2. Expected Value if B plays b1, Expected Value if B plays b2,

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**Mixed Strategies - Example**

Player A Player B b1 B2 a1 1 a2 - 4 3 Let q be the proportion of time B plays row b1 , then (1 – q) is the proportion of time that he plays b2. He would expect the same payoff irrespective of whether A plays a1 or a2. Expected Value if A plays a1, Expected Value if A plays a2,

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**Mixed Strategies - Example**

Player A Player B b1 B2 a1 1 a2 - 4 3 A should play strategy a1 for 7/8 of the time and a2 for 1/8 of the time. B should play strategy b1 for 3/8 of the time and b2 for 5/8 of the time. Value of the game is 3/8

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**Mixed Strategies – Short Cut Method**

Player B b1 b2 Player A a1 1 1 7/8 7 a2 - 4 3 7 1 1/8 3/8 3 5 5/8 5 3 Subtract the smaller payoff in each row from the larger payoff, and the smaller payoff in each column from the larger payoff. Interchange each of these pairs of subtracted numbers Divide each pair of numbers by the sum of the pair Strategy for A (7/8, 1/8) and for B (3/8, 5/8)

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Dominance In a game, it is often possible to find an entire row (or column) which one player will avoid when there is another row (or column) which is always better for him to play. In that case, we say that the row (or column) is dominated by another row (or column). If all values of elements of Row A are greater than corresponding values of Row B, we can eliminate row B as Row A dominates Row B. The Row player will never play strategy B, if he has a choice between strategy A and B as A always gives him more payoff. Similarly, if the value of elements in Column A are all smaller than corresponding values of Column B, then Column A dominates Column B and Column B can be eliminated.

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Dominance - Example Two breakfast food manufacturing firms A and B are competing for an increased market share. To improve their market share both the firms have the following options: Give coupons – a1, b1 Decrease price – a2, b2 Maintain present strategy – a3, b3 Increase advertising – a4, b4

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**b4 also dominates column b2.. Eliminate b2 **

Firm A Firm B b1 b2 b3 b4 a1 35 65 25 5 a2 30 20 15 a3 40 50 10 a4 55 60 20 5 20/25 5/25 15 10 10/25 15/25 All payoffs in the row a1 are higher than the payoffs in the row a2. a1 dominates a2. Eliminate a2 All payoffs in the row a4 are higher than the payoffs in the row a3. a4 dominates a3. Eliminate a3 All payoffs in the column b4 are lower than the payoffs in the column b1. b4 dominates b1. Eliminate b1 b4 also dominates column b2.. Eliminate b2 Solve by short cut method. A’s Strategy (1/5,0,0,4/5) B’s Strategy (0,0,2/5,3/5). Value of game 13.

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**Graphical Solution of Games**

Used for solving two person zero sum 2 X n or m X 2 games ( 2 rows and n columns or m rows and 2 columns). B b1 b2 b3 b4 A a1 7 -7 4 -8 a2 -6 2 6 Assume that Player A plays strategy a1 for p proportion of time and strategy a2 for (1 - p) part of the time, then if B plays strategy b1, A’s payoff will be: 13p – 6 Work out for other strategies of B and plot on a graph.

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**Player A’s aim is to maximise his minimum payoff**

Player A’s aim is to maximise his minimum payoff. Consider the lower shaded envelope and find the highest point, which in this case is P. OP gives the value of p = 5/12 PP’ is value of game = -7/12 P lies on the intersection of strategies b1 and b2. Solve as a 2 X 2 game O b2 b3 b4 b1 P P’ p

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**b1 b4 b3 b2 p The game reduces to the one shown above.**

7 -7 10/24 A -6 4 14/24 11/24 13/24 O b2 b3 b4 b1 P P’ p The game reduces to the one shown above. Solving by short cut method, strategy for A is (5/12,7/12) and for B (11/24,13/24,0,0) Value of the game is -7/12.

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**Graphical Solution of Games**

If the game is of the form m X 2, the procedure remains the same, except that the upper envelope is considered, as the column player is minimising his maximum loss. Player B b1 b2 a1 7 -5 Player A a2 2 6 a3 -6 3 a4 8 -1

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**OQ’ gives the value of q = 7/13 **

QQ’ gives the value of the game = 50/13 The game can be reduced to a 2 X 2 game with the strategies being a2, a4,b1 and b2. Strategies for A (0, 9/13,0, 4/13) and for B (7/13, 6/13) a2 a3 a4 a1

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Summary The game theory has too many assumptions to be of any real practical value. It can be used only in a limited context. n person games are more complex as some players may collaborate against others. There may be negotiations and so on. Solutions to such games are currently not available.

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Operations Research Assistant Professor Dr. Sana’a Wafa Al-Sayegh 2 nd Semester 2008-2009 ITGD4207 University of Palestine.

Operations Research Assistant Professor Dr. Sana’a Wafa Al-Sayegh 2 nd Semester 2008-2009 ITGD4207 University of Palestine.

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