Title: Game Theory and Strategy
1Game Theory and Strategy
2Content
- Two-persons Zero-Sum Games
- Two-Persons Non-Zero-Sum Games
- N-Persons Games
3Introduction
- At least 2 players
- Strategies
- Outcome
- Payoffs
4Two-persons Zero-Sum Games
- Payoffs of each outcome add to zero
- Pure conflict between 2 players
5Two-persons Zero-Sum Games
6Two-persons Zero-Sum Games
7Dominance and Dominance Principle
- Definition A strategy S dominates a strategy T
if every outcome in S is at least as good as the
corresponding outcome in T, and at least one
outcome in S is strictly better than the
corresponding outcome in T. - Dominance Principle A rational player would
never play a dominated strategy.
8Saddle Points and Saddle Points Principle
- Definition An outcome in a matrix game is called
a Saddle Point if the entry at that outcome is
both less than or equal to any in its row, and
greater than or equal to any entry in its column. - Saddle Point Principle If a matrix game has a
saddle point, both players should play a strategy
which contains it.
9Value
- Definition For a matrix game, if there is a
number such that player A has a strategy which
guarantees that he will win at least v and player
B has a strategy which guarantees player A will
win no more than v, then v is called the value of
the game.
10Two-persons Zero-Sum Games
11Saddle Points
12Saddle Points
- 0 saddle point
- 1 saddle point
- more than 1 saddle points
13Mixed Strategy
14Mixed Strategy
- Colin plays with probability x for A, (1-x) for B
- Rose A x(2) (1-x)(-3) -3 5x
- Rose B x(0) (1-x)(3) 3 - 3x
- if -3 5x 3 - 3x gt x 0.75
- Rose A 0.75(2) 0.25(-3) 0.75
- Rose B 0.75(0) 0.25(3) 0.75
15Mixed Strategy
- Rose plays with probability x for A, (1-x) for B
- Colin A x(2) (1-x)(0) 2x
- Colin B x(-3) (1-x)(3) 3 - 6x
- if 2x 3 - 6x gt x 0.375
- Colin A 0.375(2) 0.625(0) 0.75
- Colin B 0.375(-3) 0.625(3) 0.75
16Mixed Strategy
- 0.75 as the value of the game
- 0.75A, 0.25B as Colins optimal strategy
- 0.375A. 0.625B as Roses optimal strategy
17Mixed Strategy
18Minimax Theorem
- Every m x n matrix game has a solution. There is
a unique number v, called the value of game, and
optimal strategy for the players such that - i) player As expected payoff is no less that v,
no matter what player B does, and - ii) player Bs expected payoff is no more that v,
no matter what player A does - The solution can always be found in k x k subgame
of the original game
19Minimax Theorem (example)
20Minimax Theorem (example)
- There is no dominance in the above example
- From arrows in the graph, Colin will only choose
A, B or C, but not D or E. - So the game is reduced into a 3 x 3 subgame
21Example
22Example
23Example
24Example
25Mixed Strategy
26Utility Theory
27Utility Theory
- Roses order is u, w, x, z, y, v
- Colins order is v, y, z, x, w, u
28Utility Theory
29Utility Theory
- Transformation can be done using a positive
linear function, f(x) ax b - in this example, f(x) 0.5(x - 17)
-
--------gt
30Two-Persons Non-Zero-Sum Games
- Equilibrium outcomes in non-zero-sum games
saddle points in zero-sum games
31Prisoners Dilemma
32Nash Equilibrium
- If there is a set of strategies with the property
that no player can benefit by changing her
strategy while the other players keep their
strategies unchanged, then that set of strategies
and the corresponding payoffs constitute the Nash
Equilibrium
33Dominant Strategy Equilibrium
- If every player in the game has a dominant
strategy, and each player plays the dominant
strategy, then that combination of strategies and
the corresponding payoffs are said to constitute
the dominant strategy equilibrium for that game.
34Pareto-optimal
- If an outcome cannot be improved upon, ie. no one
can be made better off without making somebody
else worse off, then the outcome is Pareto-optimal
35Pareto Principle
- To be acceptable as a solution to a game, an
outcome should be Pareto-optimal.
36Prudential Strategy, Security Level and
Counter-Prudential Strategy
- In a non-zero-sum game, player As optimal
strategy in As game is called As prudential
strategy. - The value of As game is called As security
level - As counter-prudential strategy is As optimal
response to his opponents prudential strategy.
37Example
38Example
- consider only Roses strategy
- saddle point at AB
39Example
- consider only Colins strategy
40Example
41Example
42Example
43Co-operative Solution
44Co-operative Solution
45Co-operative Solution
- Concerns are Trust and Suspicion
46N-Person Games
- More important and common in real life
- n is assumed to be at least three
47N-Person Games
48N-Person Games
49N-Person Games
50N-Person Games
51N-Person Games
52N-Person Games
53N-Person Games
54N-Person Games
55Superaddictive
- A characteristic function form game (N, v) is
called superadditive - if v(S, T) gt v(S) v(T) for any two coalitions
S and T
56N-Person Prisoners Dilemma
57N-Person Prisoners Dilemma
- General form of N-Person Prisoners Dilemma
- each of n players has two strategies, C and D
- for every player, D is a dominant strategy
- if all players choose D, add will be worse off
than if all players had chosen C
58Example
59Example
60Example
61From the bottom up algorithm
- i) under optimal play, the Reds choice in last
round will be the player who is last on the
Blues preference list. Mark that player as the
Reds last round choice and cross him off both
teams lists - ii) the Blues choice in last round will be the
player who is last on the Reds reduced list.
Mark the player as Blues and cross him off both
teams lists - iii) continue like this, finding the choices in
the next-to-last round, and on up to the first
round
62Example
63Example
64Example
65Example
66Example of N-Person Prisoners Dilemma
67Example of N-Person Prisoners Dilemma
68Example of N-Person Prisoners Dilemma
- After Greens optimal Choice
69Example of N-Person Prisoners Dilemma
- After Reds optimal Choice
70Example of N-Person Prisoners Dilemma
- After Blues optimal Choice
71END