Game Theory
description
Transcript of Game Theory
![Page 1: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/1.jpg)
Game Theory
![Page 2: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/2.jpg)
Game TheoryTwo (or more) decision makers with
conflicting interests are under competition.
![Page 3: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/3.jpg)
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.
![Page 4: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/4.jpg)
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.
![Page 5: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/5.jpg)
Example 1 (payoff/penalty table)
Athlete Manager’s StrategiesStrategies (Column Strategies)(row strat.) A B C
1 $50,000 $35,000 $30,000 2 $60,000 $40,000 $20,000
![Page 6: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/6.jpg)
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.
![Page 7: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/7.jpg)
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.
![Page 8: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/8.jpg)
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
![Page 9: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/9.jpg)
An equivalent zero-sum table
Network 2Network 1 western Soap
Comedy
western -15 -35 10 soap - 5 8 0
comedy -12 -36 20
![Page 10: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/10.jpg)
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.
![Page 11: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/11.jpg)
Example 3 (equilibrium point)
Athlete Manager’s StrategiesStrategies (Column Strategies)(row strat.) A B C
1 $50,000 $35,000 $30,000 2 $60,000 $40,000 $20,000
![Page 12: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/12.jpg)
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.
![Page 13: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/13.jpg)
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.
![Page 14: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/14.jpg)
Pure Strategy of Example 3
Athlete Manager’s StrategiesStrategies (Column Strategies)(row strat.) A B C
1 $50,000 $35,000 $30,000 2 $60,000 $40,000 $20,000
![Page 15: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/15.jpg)
Example 4 (2-person, 0-sum)
RowPlayers Column Player Strategies Strategies 1 2 3
1 4 4 10 2 2 3 1 3 6 5 7
![Page 16: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/16.jpg)
Mixed StrategyIf a game does not have an
equilibrium, the best strategy would be a mixed strategy.
![Page 17: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/17.jpg)
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.
![Page 18: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/18.jpg)
Example:
Company I Company II Strategies Strategies B C
2 8 43 1 7
![Page 19: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/19.jpg)
Mixed Strategy
A mixed strategy for a player is a set of probabilities each for an alternative strategy of the player.
![Page 20: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/20.jpg)
Example: Mixed Strategy
Company I Company II Strategies Strategies B C
2 8 43 1 7
Let mixed strategy for company I be{0.6, 0.4}; and for Company II be{0.3, 0.7}.
![Page 21: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/21.jpg)
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.
![Page 22: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/22.jpg)
Expected Value
The outcome X of something is uncertain, which can be X1, X2, …, Xn with probabilities p1, p2, …, pn respectively.
Expected value of X is:E(X) = X1*p1+X2*p2+…+Xn*pn
E(X) is interpreted as the average outcome.
![Page 23: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/23.jpg)
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:
![Page 24: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/24.jpg)
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:
![Page 25: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/25.jpg)
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).
![Page 26: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/26.jpg)
How to Find Equilibrium Mixed Strategy
By linear programming (as introduced in book)
By QM for Windows, – we use this approach.
![Page 27: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/27.jpg)
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.
![Page 28: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/28.jpg)
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).
![Page 29: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/29.jpg)
Unilateral Change from Equilibrium by Column Playerprobability 0.1 0.9
B C0.6 Strat 2 8 40.4 Strat 3 1 7
![Page 30: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/30.jpg)
Unilateral Change from Equilibrium by Column Playerprobability 1.0 0
B C0.6 Strat 2 8 40.4 Strat 3 1 7
![Page 31: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/31.jpg)
Unilateral Change from Equilibrium by Row Player
probability 0.3 0.7 B C
0.2 Strat 2 8 40.8 Strat 3 1 7
![Page 32: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/32.jpg)
A Double-Secure StrategyAt 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)
![Page 33: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/33.jpg)
Both Leave Their Equilibrium Strategies
probability 0.8 0.2 B C
0.5 Strat 2 8 40.5 Strat 3 1 7
![Page 34: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/34.jpg)
Both Leave Their Equilibrium Strategies
probability 0 1 B C
0.2 Strat 2 8 40.8 Strat 3 1 7
![Page 35: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/35.jpg)
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.
![Page 36: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/36.jpg)
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 (!).
![Page 37: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/37.jpg)
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.
![Page 38: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/38.jpg)
Example:
Company I Company II Strategies Strategies A B C
1 9 7 22 11 8 43 4 1 7
![Page 39: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/39.jpg)
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)
![Page 40: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/40.jpg)
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
![Page 41: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/41.jpg)
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.
![Page 42: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/42.jpg)
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.
![Page 43: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/43.jpg)
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.
![Page 44: Game Theory](https://reader035.fdocuments.in/reader035/viewer/2022062810/56815e01550346895dcc45c0/html5/thumbnails/44.jpg)
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.