Game Theory

Game Theory Two (or more) decision makers with conflicting interests are under competition.

Zero-Sum vs. Non-zero-Sum In a zero-sum game, a player’s gain is equal to another player’s loss. In a non-zero-sum game, a player’s gain is not necessarily equal to another player’s loss.

Two-Person Zero-Sum Game Two decision makers’ benefits are completely opposite i.e., one person’s gain is another person’s loss Payoff/penalty table (zero-sum table): shows “offensive” strategies (in rows) versus “defensive” strategies (in columns); gives the gain of row player (loss of column player), of each possible strategy encounter.

Example 1 (payoff/penalty table) Athlete Manager’s Strategies Strategies (Column Strategies) (row strat.) A B C 1 $50,000 $35,000 $30,000 2 $60,000 $40,000 $20,000

Two-Person Constant-Sum Game For any strategy encounter, the row player’s payoff and the column player’s payoff add up to a constant C. It can be converted to a two-person zero-sum game by subtracting half of the constant (i.e. 0.5C) from each payoff.

Example 2 (2-person, constant-sum) During the 8-9pm time slot, two broadcasting networks are vying for an audience of 100 million viewers, who would watch either of the two networks.

Payoffs of NW1 for the constant-sum of 100(million) Network 1 Network 2 (NW2) (NW1) western Soap Comedy western 35 15 60 soap 45 58 50 comedy 38 14 70

An equivalent zero-sum table Network 2 Network 1 western Soap Comedy western -15 -35 10 soap - 5 8 0 comedy -12 -36 20

Equilibrium Point In a two-person zero-sum game, if there is a payoff value P such that P = max{row minimums} = min{column maximums} then P is called the equilibrium point, or saddle point, of the game.

Example 3 (equilibrium point) Athlete Manager’s Strategies Strategies (Column Strategies) (row strat.) A B C 1 $50,000 $35,000 $30,000 2 $60,000 $40,000 $20,000

Game with an Equilibrium Point: Pure Strategy The equilibrium point is the only rational outcome of this game; and its corresponding strategies for the two sides are their best choices, called pure strategy. The value at the equilibrium point is called the value of the game. At the equilibrium point, neither side can benefit from a unilateral change in strategy.

Equilibrium A status is equilibrium if it is balanced; and once it is off the balance, it will be brought back to balance automatically by its internal force.

Pure Strategy of Example 3 Athlete Manager’s Strategies Strategies (Column Strategies) (row strat.) A B C 1 $50,000 $35,000 $30,000 2 $60,000 $40,000 $20,000

Example 4 (2-person, 0-sum) Row Players Column Player Strategies Strategies 1 2 3 1 4 4 10 2 2 3 1 3 6 5 7

Mixed Strategy If a game does not have an equilibrium, the best strategy would be a mixed strategy.

Game without an Equilibrium Point max{row minimums} ≠ min{column maximums} At least one player may benefit from unilateral change from any strategy. So, the game would get into a loop. To break loop, a mixed strategy is applied.

Example: Company I Company II Strategies Strategies B C 2 8 4 3 1 7 2 8 4 3 1 7 Starting from any strategy in a “simpliest” payoff table, (2,C) for example. The row player would change to stra. 3 to make more benefit. The column player finds that, and will change to strategy B to reduce its loss to 1. Then, row would change to stra. 2 to increase benefit to 8. Then column would change back to strat. C to reduce loss to 4. Thus complete a loop.

Mixed Strategy A mixed strategy for a player is a set of probabilities each for an alternative strategy of the player.

Example: Mixed Strategy Company I Company II Strategies Strategies B C 2 8 4 3 1 7 Let mixed strategy for company I be {0.6, 0.4}; and for Company II be {0.3, 0.7}. E(strat 2) = 8*0.3+4*0.7 = 5.2 E(strat 3) = 1*0.3+7*0.7 = 5.2 E(row player) = 5.2*0.6+5.2*0.4=5.2 E(strat B) = 8*0.6+1*0.4 = 5.2 E(strat C) = 4*0.6+7*0.4 = 5.2 E(column player) = 5.2*0.3+5.2*0.7=5.2 When row player stays with his equilibrium strategy {0.6, 0.4}, expected values of all column strategies are same. Therefore, no matter what strategy the column employs, the row player’s expected gain would remain unchanged.

Equilibrium Mixed Strategy An equilibrium mixed strategy makes expected values of any player’s individual strategies identical. Every game contains one equilibrium mixed strategy. The equilibrium mixed strategy is the best strategy for both.

Expected Value The outcome of an action A is uncertain. X1, X2, …, Xn are possible outcomes with probabilities p1, p2, …, pn respectively. Expected value of A’s outcomes is: E(A) = X1*p1+X2*p2+…+Xn*pn E(A) is interpreted as the average outcome of action A.

Example (continued) If Company I takes strategy 2, then the expected value of payoffs would be: If Company I takes strategy 3, then the expected value of payoffs would be: Expected value of payoffs for Company I is:

Example (continued) If Company II takes strategy B, then the expected value of losses would be: If Company II takes strategy C, then the expected value of losses would be: Expected value of losses for Company II is:

Pure Strategy Is a Special Case of Mixed Strategy If a probability in a mixed strategy equals to 1, then it becomes a pure strategy. An equilibrium mixed strategy, say, (0, 1, 0) for row player , (1, 0) for column player , is a pure equilibrium strategy: (strategy 2 for row player, strategy 1 for column player).

How to Find Equilibrium Mixed Strategy By linear programming (as introduced in book) By QM for Windows, – we use this approach.

Both Are Better Off at Equilibrium At equilibrium, both players are better off, compared to maximin strategy for row player and minimax strategy for column player. No player would benefit from unilaterally changing the strategy.

A Care-Free Strategy The row player’s expected gain remains constant as far as he stays with his mixed strategy (no matter what strategy the column player uses). The column player’s expected loss remains constant as far as he stays with his mixed strategy (no matter what strategy the row player uses).

Unilateral Change from Equilibrium by Column Player probability 0.1 0.9 B C 0.6 Strat 2 8 4 0.4 Strat 3 1 7 E(strat 2) = 0.8+3.6 = 4.4 E(strat 3) = 0.1+6.3 = 6.4 E(row player) = 4.4*0.6+6.4*0.4=5.2 = E(column player) E(row player) = E(column player) = 5.2

Unilateral Change from Equilibrium by Column Player probability 1.0 0 B C 0.6 Strat 2 8 4 0.4 Strat 3 1 7 E(row player) = E(column player) = 5.2

Unilateral Change from Equilibrium by Row Player probability 0.3 0.7 B C 0.2 Strat 2 8 4 0.8 Strat 3 1 7 E(strat 2) = 2.4+2.8=5.2 E(strat 3) = 0.3+4.9=5.2 E(row player) = 5.2*0.2+5.2*0.8=5.2 = E(column player)

A Double-Secure Strategy At the equilibrium, the expected gain or loss will not change unless both players give up their equilibrium strategies. Note: Expected gain of row player is always equal to expected loss of column player, even not at the equilibrium, since 0-sum)

Both Leave Their Equilibrium Strategies probability 0.8 0.2 B C 0.5 Strat 2 8 4 0.5 Strat 3 1 7 E(strat 2) = 6.4+0.8=7.2 E(strat 3) = 0.8+1.4=2.2 E(row player) = 7.2*0.5+2.2*0.5=4.7 E(strat B) = 4+0.5=4.5 E(strat C) = 2+3.5=5.5 E(column player) = 4.5*0.8+5.5*0.2=4.7 This example shows, row player is worse off and column player is better off when they both leave the equilibrium.

Both Leave Their Equilibrium Strategies probability 0 1 B C 0.2 Strat 2 8 4 0.8 Strat 3 1 7 E(strat 2) = 4 E(strat 3) = 7 E(row player) = 4*0.2+7*0.8 = 0.8+5.6 = 6.4 E(strat B) = 1.6+0.8 = 2.4 E(strat C) = 0.8+5.6 = 6.4 E(column player) = 2.4*0+6.4*1 = 6.4 This example shows that row player is better off, but column player is worse off, if they both leave the equilibrium.

Penalty for Leaving Equilibrium It is equilibrium because it discourages any unilateral change. If a player unilaterally leaves the equilibrium strategy, then his expected gain or loss would not change, and once the change is identified by the competitor, the competitor can easily beat the non-equilibrium strategy.

Implementation of a Mixed Strategy Applied in the situations where the mixed strategy would be used many times. Randomly select a strategy each time according to the probabilities in the strategy. If you had good information about the payoff table, you could figure out not only your best strategy, but also the best strategy of your competitor (!).

Dominating Strategy vs. Dominated Strategy For row strategies A and B: If A has a better (larger) payoff than B for any column strategy, then B is dominated by A. For column strategies X and Y: if X has a better (smaller) payoff than Y for any row strategy, then Y is dominated by X. A dominated decision can be removed from the payoff table to simplify the problem.

Example: Company I Company II Strategies Strategies A B C 1 9 7 2 1 9 7 2 2 11 8 4 3 4 1 7

2-Person Non-zero Sum Game One player’s gain is not equal to the other player’s loss. Prisoners’ dilemma (see a separate PowerPoint presentation)

Decision Theory Problems If one of players in a game theory problem is the “Mother Nature” or the “God”, then it becomes a Decision Theory Problem. For example: To determine which stock would be selected for your investment; Hoe many cases of milk to order every week for a grocery store; How many cashiers to hire

Examples of Decision Theory Problems How high the dam should be built to deal with possible flood; Which stock would be selected for your investment; How many cases of milk to order every week for a grocery store; How many cashiers to hire to serve customers at a satisfactory level.

Major Difference In a game theory problem, one player’s strategy would affect the other’s strategy. In a decision theory problem, the action of Mother Nature is not influenced by a human’s. Mother Nature’s action is simply random in the eyes of humans, which is called state of nature.

Approaches for Making Decision (1) If probabilities of states of nature can be figured out, then the alternative with highest expected value of possible payoffs will be the best decision, by using a decision table or a decision tree.

Approaches for Making Decision (2) If probabilities of states of nature are not known, then there are a couple of criteria to make decision, dependent on decision maker’s preference, as studied in Cht. 12.