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Nash Equilibrium

- A strategy profile s (s1, , sn) is a Nash

Equilibrium if, for all agents i, si is a best

response to s-i. - two strategies si and s-i are in Nash equilibrium

if - under the assumption that agent i plays si, agent

j can do no better than play s-i and - under the assumption that agent j plays s-i,

agent i can do no better than play si. - Each agent chooses a strategy that is a best

response to the other agents strategies.

Nash Equilibrium (NE)

- How hard is the Nash Equilibrium to compute?

Normal Form Games

- Finite, n-person normal form game (N, A, O, m,

u) - N is a finite set of n players, indexed by i
- A A1, . . . , An is a set of actions for each

player i - a A is an action profile
- O is a set of outcomes
- m A O
- u u1, ... , un, a utility function for each

player, where ui A

Game Definitions

- Common payoff games a game where all action

profiles a A1 x x An for any pair of agents i,

j, result in ui(a) uj(a). - Constant sum games A game in which a constant c

exists st for each strategy profile a A1 x A2 it

is the case that u1(a) u2(a) c.

Linear Programming

- A linear program is defined by
- A set of real-valued variables
- A linear objective function
- A weighted sum of the variables
- A set of linear constraints
- The requirement that a weighted sum of the

variables must be greater than or equal to some

constant

Linear programming

- maximize Si wixi
- subject to Si wcixi bc c C
- xi 0 xi X
- These problems can be solved in polynomial time

using interior point methods. - Interestingly, the (worst-case exponential)

simplex method is often faster in practice.

NE of two-player zero-sum games

- Easiest NE to find via linear programming in

polynomial time. - Let G (1,2, A1, A2, u1, u2) be a two-player

zero-sum game. - Ui is the unique expected utility for player i

in equilibrium. - We know that the players interests are opposing,

what does this tell us about U1 and U2? - U1 - U2

NE of two-player zero-sum games

- minimize U1
- subject to u1(aj1, ak2) sk2 U1

j A1 - sk2 1
- sk2 0 k A2
- Variables
- U1 is the expected utility for player 1
- sa22 is player 2s probability of playing action

a2 under his mixed strategy - each u1(a1, a2) is a constant.

NE of two-player zero-sum games

- minimize U1
- subject to u1(aj1, ak2) sk2 rj1 U1

j A1 - sa22 1
- sk2 0 k A2
- rj1 0 j A1
- Introduce slack variables, rj1.

NE of two-player general-sum games

- The NE of a two-player general-sum game cannot be

represented by a linear program. Why?

NE of two-player general-sum games

- Computation of the NE in this case is not

NP-Complete. Why? - However, it does appear to be in the PPAD

complexity class.

NE of two-player general-sum games

- PPAD Class of problems
- Define a family of directed graphs, G(n).
- Let each graph in G(n) contain a number of nodes

that is exponential in n. - Let Parent N N and Child N N
- Let there be one graph g G(n) for every such

pair of Parent and Child functions as long as - An edge exists from a node j to a node k iff

Parent(k) j and Child(j) k. - There must exist one distinguished node 0 N

with exactly zero parents.

NE of two-player general-sum games

- Reformulate previous solution to explicitly

consider both players. - u1(aj1, ak2) sk2 rj1 U1 j A1
- u2(aj1, ak2) sk1 rj2 U2 k A1
- sj1 1, sk2 1
- sj1 0, sk2 0 j A1, k A2
- rj1 0, rk2 0 j A1 , k A2
- rj1 sj1 0, rk2 sk2 0 j A1 , k A2

NE of two-player general-sum games

- The complementarity condition requires that when

an action is played by a particular player with

positive probability, the corresponding slack

variable must be zero. - Each slack variable represents the players

incentive to deviate from the corresponding

action. - Linear Complementarity Problem (LCP)

NE of two-player general-sum games

- Solve the LCP with the Lemke-Howson algorithm

a12

a22

How many pure strategies does agent a1 have?

a11

What about agent a2?

a21

a31

NE of two-player general-sum games

- Lemke-Howson algorithm

a12

a22

a11

a21

a31

How do we represent a2s Strategy space?

How do we represent a1s Strategy space?

NE of two-player general-sum games

- Lemke-Howson algorithm
- Label the strategies.
- A pair of strategies (s1, s2) is a NE iff it is

completely labeled (L(s1) L(s2) A1 A2).

s31

s22

Where are the multiply labeled points?

(0,0,1)

(0,1)

(0,1/3,2/3)

(1/3, 2/3)

(1,0,0)

s11

(2/3, 1/3)

s12

(2/3,1/3,0)

(1,0)

(0,1,0)

s21

NE of two-player general-sum games

- Lemke-Howson algorithm
- Identify the NE

(0,0,1)

a11 , a21 , a12

(0,1)

a11 , a12

(0,1/3,2/3)

(1/3, 2/3)

a11 , a12 , a22

a11 , a21

a21 , a31 , a12

(1,0,0)

(2/3, 1/3)

(0,0,0)

a21 , a31

a11 , a21 , a31

(2/3,1/3,0)

a31 , a12 , a22

(0,0)

(1,0)

a31 , a22

a11 , a22

(0,1,0)

a11 , a31 , a22

((0,0,1), (0,1)), ((0,1/3,2/3), (2/3,

1/3)), ((2/3,1/3,0), (1/3, 2/3))

NE of two-player general-sum games

- Lemke-Howson algorithm
- Initialize
- (s1, s2) (0, 0)
- Find an s1 G1 s.t. s1 is adjacent to 0
- x 1
- Repeat
- sx sx
- let aji be the label that occurs in both s1 and

s2 - Find an sx Gx s.t. sx is adjacent to sx and

aji L(sx) - x 3 x
- Until (s1, s2) is a completely labeled pair.

NE of two-player general-sum games

(0,0,1)

(0,0,1)

a11 , a21 , a12

a11 , a21 , a12

(0,1)

(0,1)

a11 , a12

a11 , a12

(0,1/3,2/3)

(0,1/3,2/3)

(1/3, 2/3)

(1/3, 2/3)

(1/3, 2/3)

a11 , a12 , a22

a11 , a12 , a22

a11 , a21

a11 , a21

a11 , a21

a21 , a31 , a12

(1,0,0)

(2/3, 1/3)

(2/3, 1/3)

(2/3, 1/3)

(0,0,0)

(0,0,0)

a21 , a31

a21 , a31

a21 , a31

a11 , a21 , a31

a11 , a21 , a31

(2/3,1/3,0)

(2/3,1/3,0)

a31 , a12 , a22

a31 , a12 , a22

(0,0)

(1,0)

(1,0)

(0,0)

(1,0)

a31 , a22

a31 , a22

a31 , a22

a11 , a22

a11 , a22

(0,1,0)

(0,1,0)

a11 , a31 , a22

a11 , a31 , a22

((2/3,1/3,0), (1/3,2/3)) ((0,0,0), (0,0))

((0,1,0), (0,1)) ((2/3,1/3,0), (0,1))

((2/3,1/3,0), (1/3,2/3))

((2/3,1/3,0), (1/3,2/3)) ((2/3,1/3,0),

(1/3,2/3)) ((0,1/3,2/3)), (1/3,2/3)

((0,1/3,2/3), (2/3, 1/3))

((0,0,1), (1,0)) ((0,0,0), (0,0)) ((0,0,1)),

(0,0) ((0,0,1), (1,0))

NE of two-player general-sum games

- Lemke-Howson algorithm
- Advantages
- Guaranteed to find a sample NE.
- Non-determinism is concentrated in the first

move. - Disadvantages
- Not guaranteed to find all NE.
- Does not provide guidance on choosing a good

first move.

NE of two-player general-sum games

- Heuristics and the Support-Enumeration Method can

be combined to provide an algorithm to find NE. - Search for NE can be reduced to searching the

space of supports. - Use a feasibility program that tests the

specified supports. - Complete
- Worst case performance exponential

Specific NE of General-sum Games

- Idea is to find an equilibrium with a specific

property. - Properties
- Uniqueness
- Pareto optimal
- Guaranteed payoff
- Guaranteed social welfare
- Subset inclusion
- Subset containment
- NP-Hard when applied to NE.

NE for n-player general-sum games

- Cannot formulate as a linear complementarity

problem. - Sequence of linear complementarity problems

(SLCP). - Each LCP is an approximation of the problem and

is used to developed the next approximation in

the sequence.

NE for n-player general-sum games

- Formulate the problem as a minimum of a function.
- Constrained optimization problem
- Unconstrained optimization problem
- Disadvantages
- Both have local minima that do not correspond to

the NE.

NE for n-player general-sum games

- Simplicial subdivision algorithms
- Consider
- the space of mixed strategies is a simlpex
- The players best response is a function from

points on the simplex to other points on the

simplex. - Scarfs algorithm locates the fixed points.
- Add a variable that expresses the accuracy of the

current iterations approximation. - Worst case complexity exponential in the number

of players and the number of digits of accuracy.

NE for n-player general-sum games

- Generalize SEM to the n-player case
- The feasibility program becomes non-linear.
- Algorithm must accommodate multiple variables in

the feasibility problem. - Use standard numerical techniques for non-linear

optimization. - Reverse the lexicographic ordering between size

and balance of supports.

All NE of General-sum Games

- Idea is to determine all equilibria of a game.
- Important when designing a game and need to know

all possible stable outcomes. - Worst case exponential in the number of actions

for each player.

Dominant Strategies

- A strategy dominates another when the first

strategy is always at least as good as the

second, independent of the other players

actions. - Iterative removal
- Strictly dominant strategies order does not

matter. - Very weakly and weakly dominant strategies

removal order can have an affect. - Potentially remove some equilibrium of the

original game. - Potentially remove a larger set of strategies and

result in a smaller game.

Domination by a Pure Strategy

- for all pure strategies ai Ai for player i where

ai ? si do - dom true
- for all pure strategy profiles a-i A-i for the

players other than i do - if ui(si, a-i) ui(ai, a-i) then
- dom false
- break
- end if
- end for
- if dom true then return true
- end for
- return false

Domination by a Mixed Strategy

- Recall that mixed strategies cannot be

enumerated. - Strict Domination
- Requires a linear program.
- Minimize
- Subject to

Domination by a Mixed Strategy

- Very weakly dominate

Domination by a Mixed Strategy

- Weak domination

Maximize

Subject to

Iterated Dominance

- Strategy Elimination Does there exist some

elimination path under which the strategy si is

eliminated? - Reduction Identity Given action subsets A-i Ai

for each player i, does there exist a maximally

reduced game where each player i has the actions

A-i? - Uniqueness Does every elimination path lead to

the same reduced game? - Reduction Size Given constants ki for each

player i, does there exist a maximally reduced

game where each player i has exactly ki actions? - Iterated strict dominance problems are all in P.
- Iterated weak or very weak dominance problems

are NP-complete.

Correlated Equilibrium n-player general-sum games

Variables p(a) constants ui(a)

Could find the social-welfare maximizing the

correlated equilibrium by adding an objective

function

Maximize

Correlated Equilibrium

- Complexity of P when applied to CE
- Uniqueness
- Pareto Optimal
- Guaranteed payoff
- Subset inclusion
- Subset containment

CE to NE Calculation

Intuitively, correlated equilibrium has only a

single randomization over outcomes, whereas in NE

this is constructed as a product of independent

probabilities.

Changing this program so that it finds NE

requires the first constraint to be

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