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Computational Issues in

Game Theory Lecture 1

Matrix Games

- Edith Elkind
- Intelligence, Agents, Multimedia group (IAM)
- School of Electronics and CS
- U. of Southampton

Games and Strategies

- Games strategic interactions between rational

entities - Solution concepts whats going to happen?
- dominant strategies
- Nash equilibrium
- .
- Can it be computed?
- if your computer cannot find it, the market

probably cannot either

Matrix (Normal Form) Games

- finite set of players 1, , n
- each player has k actions
- pure strategies actions 1, , k
- mixed strategies probability dists over actions

- payoffs of the ith player Pi 1, , kn ? R

Row player

Column player

Prisoners Dilemma

- C collaborate, D defect
- for each player D is better than C
- no matter what the other player does
- D is a dominant strategy

Row player

Column player

Dominant Strategy Definition

- Notation
- si strategy of player i
- s-i (s1, , si-1, si1, , sn)
- s, t two strategies of player i
- s strictly dominates t if for any s-i Pi (s, s-i

) gt Pi (t, s-i ) - s weakly dominates t if for any s-i Pi (s, s-i )

Pi (t, s-i ) and ? s-i s.t. Pi

(s, s-i ) gt Pi (t, s-i ) - s is (weakly/strictly) dominant for i if

it

(weakly/strictly) dominates all t ? s

Dominant Strategy Discussion

- Very strong solution concept
- no assumptions about other players
- may not exist
- e.g., co-ordination game

Row player

Column player

Eliminating Dominated Strategies

- two players X and Y, strategies x and y
- 10 strategies per player 1, , 10
- X gets 1 if x is closer to 0.9(xy)/2 than y,

0 otherwise - 9 dominates 10 0.9(xy)/2 9
- if 10 is eliminated, 8 dominates 9 0.9(xy)/2

.81 - if 10, 9 are eliminated, 7 dominates 8
- eventually, 1 is the only strategy not eliminated
- note 1 is not a dominant strategy!

Eliminating Dominated Strategies Discussion

- may end up with more than one strategy per player
- surviving strategy need not be dominant
- what if there is more than one dominated

strategy? - if strongly dominated, final outcome is

path-independent - if weakly dominated, may depend on choices

Nash Equilibrium

- Nash equilibrium a strategy profile such that
- noone wants to deviate given other players

strategies, i.e., each players strategy is a

best response to others strategies - Battle of Sexes (0, 0) and (1, 1) are both NE

Row player

Column player

Pure vs. Mixed Strategies

- NE in pure strategies may not exist!
- matching pennies
- Mixed strategy a probability distribution over

actions - 50 tail, 50 head

Row player

Column player

Existence of NE

- Theorem (Nash 1951)

any n-player k-action game

in normal form has an equilibrium

in mixed strategies - can we find one in efficiently?

Existence of NE proof sketch

- Brouwers Theorem Any continuous mapping from

the simplex to itself has a

fixpoint. - Nash ? Brouwer proof sketch
- set of all strategy profiles ? simplex
- mapping (s1, , sn) ? (s1d1, , sndn), where

di is a shift in the direction of

best response to (s1, , si-1, si1, , sn) - NE is a point where noone wants to deviate, i.e.,

a fixpoint

2 (rconst) players, n actions

- Input representation
- 2 players two n x n matrices
- r players r n x n x x n matrices
- poly-size for constant r
- Output representation
- for 2 players all NE are in Q
- but not for 3 and more players
- Checking for pure NE easy
- at most n2 strategy profiles

Warm-up 2-player 2-action games

Row player

Column player

BR(C)

Suppose R plays 1 w.p. r EP(C) from playing 0

(1-r)1 EP(C) from playing 1 r3 1-r gt 3r

iff r lt ¼

Suppose C plays 1 w.p. c EP(R) from playing 0

(1-c)2 EP(R) from playing 1 c1 (1-c)2 gt c

iff c lt 2/3

c

1

r

1

mixed NE r1/4, c2/3

Mixed strategies and payoffs

- Payoff matrices
- the row player plays a (a1, , an)
- the column player plays b (b1, , bn)
- expected payoff of R when playing i (Ri, , b)
- expected payoff of C when playing j (C, j, a)

R11 R12 R1n R21 R22 R2n Rn1

Rn2 Rnn

C11 C12 C1n C21 C22 C2n Cn1

Cn2 Cnn

R

C

Special case zero-sum games

- Definition a game is zero-sum if Rij -Cij
- one players gain is the others loss
- matching pennies
- Solvable in polynomial time via LP duality
- fix b row players goal

max (a, Rb) subject to ai

0, a1an 1 - dual LP

min u

subject to u (Rb)i for all i - i.e., row player is guaranteed max Rbi
- column players goal

min v subject to

v (Rb)i for all i, bi 0, b1bn 1

General case support guessing

- if 1st players strategy a supported on I ? N

ai ? 0 iff i ? I - 2nd players strategy b supported on J ?

N bj ? 0 iff j ? J - then I ? BR(b) (b, Ri, ) (b, Rk, ) for all

i? I, k? N - J ? BR(a) (a, C, j) (a, C, k)

for all j? J, k? N - LP on variables a1, , an, b1, , bn
- solutions to LP ? Nash equilibria
- guess supports, solve LP 22npoly(n) steps

linear inequalities!

Support guessing remarks

- can eliminate dominated strategies first
- strictly dominated strategies cannot be

in the support of NE - for any weakly dominated strategy, there is a NE

that does not have it in its support - may be able to reduce the problem size

considerably

Finding mixed NE

other approaches

- Naïve approaches exp(n)
- Simplex-like approach

(Lemke-Howson algorithm) - works well in practice
- exp(n) in the worst case (2004)
- Is it time to give up?
- maybe the problem is NP-hard?

Is Finding NE NP-hard?

- Reminder a problem P is NP-hard if you can

reduce 3-SAT to it - yes-instance 3-SAT ? yes-instance of P
- no-instance 3-SAT ? no-instance of P
- Problem each instance of NASH is

a yes-instance! - every game has a NE
- Formally if NASH is NP-hard then NP coNP
- Need complexity theory for

total search problems

Reducibility Among Search Problems

S X Y

T X Y

- S associates x in X with a solution set S(x)
- Total search problem for any x, S(x) is not empty

If T is easy, so is S

END OF THE LINE

- Input Boolean circuits

S (Successor), P

(Predecessor) - n inputs, n outputs
- S(0n) ? 0n, P(0n) 0n
- Output x ? 0n s.t.
- S(P(x)) ? x or P(S(x)) ? x
- Intuition G(V, E)
- V Sn
- E (x,y) yS(x), xP(y)

00000

11001

01011

01011

PPAD

- PPAD Polynomial Parity Argument, Directed

version - PPAD is the class of all search problems that are

reducible to END OF THE LINE

search problem solution

g

f

circuits S, T end of the

line

Problems in PPAD

- In PPAD
- end of the line (by definition)
- finding a fixpoint (1991)
- finding NE for rconst players (1991)
- PPAD-complete
- end of the line (by definition)
- finding a fixpoint (1991)
- finding NE
- 4 players Aug 2005, 3 players Sep 2005,

2 players Oct 2005

Approximate NE

- e-Nash equilibrium a strategy profile such that

noone can gain gt e by deviating - normalize game so that all payoffs are in 0, 1
- 2-player games
- PPAD-complete for eO(1/n)
- e 0.5
- Y starts with arbitrary y
- X sets x to be the best response to y
- Y sets z to be the best response to y and plays

(zy)/2 - current best e 0.339 (Dec 2007)

Correlated equilibrium

- Suppose that there is a central authority that

can tell each player what to do - Suppose also central authority can toss a coin
- Battle of sexes couple can achieve a fair

outcome - (1, 1) w.p. ½, (0, 0) w.p. ½
- if you are told to play 1, it is in your best

interest to do so - practical implementation theater if rains,

football if sunny

Correlated equilibrium definition

- CE dist X on the space of strategy profiles
- (NE for each player, dist on his strategies)
- s.t. conditioned on the ith component of a

profile drawn from X being s, i prefers s to

any other strategy - need not see other players signals
- CE always exist (each NE is a CE)
- CE are poly-time computable (2005)

What is a good NE?

Row player

Column player

- Nash equilibria
- (0, 0) total payoff is 3
- (1, 1) total payoff is 4
- (1/4, 2/3) total payoff is 17/12
- not all NE are created equal

Finding good NE

- checking for NE with total payoff gt T

NP-hard

maximizing individual players

payoff in a NE

NP-hard - deciding whether a particular strategy is played

in a NE NP-hard - checking if a NE is unique

NP-hard

(Gilboa, Zemel89, Conitzer, Sandholm03)

n players representation

- even with 2 strategies per player, need to

represent payoffs to each player for every action

profile (vector in 0, 1n)

n2n numbers - interesting special cases
- graphical games
- anonymous games

Graphical games

- players are vertices of a graph
- Vs payoff depends on

actions of W in N(V) U V - n players, max degree d gt

n2d1 numbers

t0, u0, v0, w0 12 t1, u0, v0, w0 31

. t1, u1, v1, w1 -6

W

Ws payoffs (16 cases)

T

V

U

Graphical games algorithms

- Graphs of max deg2

(collections of paths and cycles) - poly-time algorithm (Elkind, Goldberg, Goldberg,

ACM EC06) - Bounded-degree trees
- Exp-time algorithm/poly-time approximation

algorithm to find all NE (Kearns, Littman, Singh,

UAI 2001) - Heuristics for graphs with cycles

Graphical games hardness results

- NP-hard?
- no total search problem
- PPAD-hard?
- yes!
- in fact, this is how the hardness result for

4-player games was obtained

(Goldberg, Papadimitriou, Aug 2005)

Anonymous games

- Each players utility depends on how many other

players chose each strategy - other players are indistinguishable
- Special cases
- symmetric games in addition, all players have

the same utility function - congestion games players only care how many

other players use the same strategy - intuition strategy resource

Other concepts

- Sequential games
- players take turns to choose their moves
- solution concept subgame-perfect equilibrium
- Repeated games
- if the same game is played repeatedly, new

equilibria arise - tit-for-tat in prisoners dilemma

Summary

- Matrix games
- Solution concepts
- dominant strategies
- elimination of dominated strategies
- Nash equilibrium
- pure
- mixed
- e-Nash equilibrium
- correlated equilibrium
- Games with n players, sequential games

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