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 oppositei.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; andonce it is off the balance, it will be broughtback to balance automatically by itsinternal 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

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}.

Equilibrium Mixed StrategyAn 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 Playerprobability 0.1 0.9

B C

0.6 Strat 2 8 4

0.4 Strat 3 1 7

Unilateral Change from Equilibrium by Column Playerprobability 1.0 0

B C

0.6 Strat 2 8 4

0.4 Strat 3 1 7

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

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

Both Leave Their Equilibrium Strategies

probability 0 1

B C

0.2 Strat 2 8 4

0.8 Strat 3 1 7

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

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.