game theory
DESCRIPTION
TRANSCRIPT
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Introduction
Game theory can be defined as the study of mathematical models of conflict and cooperation between intelligent rational decision-makers.
R. Myerson
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Rationality and Intelligence
• A decision-maker is rational if he makes decisions consistently in pursuit of his own objectives
• A player is intelligent if he knows everything that we know about the game and he can make inferences about the situation that we can make
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Outcomes• Let O be a finite set of outcomes
• A lottery is a probability distribution over O
• We assume agents can rank outcomes and lotteries with a utility function
l = [p1 :o1,…, pk :ok ]
oi ∈O pi ∈[0,1]
pi = 1i=1
k
∑
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TCP GameC D
C
D
-1,-1 -4,0
0,-4 -3,-3
D: defective implementationC: correct implementation
Prisoner’s Dilemma
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Game in Normal Form• A finite n-person normal form game is a
tuple (N, A, u) where
• N is a finite set of n players
• A=A1⨉...⨉An, where Ai is a finite set of actions available to player i
• u=(u1,...,un), where ui:A↦R is a real valued utility function
• a=(a1,...,an) is an action profile
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C D
C
D
-1,-1 -4,0
0,-4 -3,-3A1
TCP Game (again)• N={1,2}
• A={C,D}⨉{C,D}
A1 A2 u1 u2
C C -1 -1C D -4 0D C 0 -4D D -3 -3
A2
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Actions• Actions can be “arbitrarily
complex”
• Ex.: international draughts
• an action is not a move
• an action maps every possible board configuration to the move to be played if the configuration occurs 2·1022
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Matching Pennies
Heads Tails
Heads
Tails
1,-1 -1,1
-1,1 1,-1
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Rock-Paper-Scissor
Rock Paper Scissors
Rock
Paper
Scissors
0,0 -1,1 1,-1
1,-1 0,0 -1,1
-1,1 1,-1 0,0
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Strategy• A pure strategy is selecting an action and playing it
• A mixed strategy for player i is an element of the set Si=Π(Ai) of probability distributions over Ai
• The support of a mixed strategy is the set of pure strategies {ai|si(ai) > 0}
• The set of mixed-strategy profiles is S1⨉...⨉Sn and a mixed strategy profile is a tuple (s1, ..., sn)
• The utility of a mixed strategy profile is
ui (s) = ui (a) s j (aj )j=1
n
∏a∈A∑
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Solution Concept
• Games are complex, the environment can be stochastic, other player’s choices affect the outcome
• Game theorists study certain subsets of outcomes that are interesting in one sense or another which are called solution concepts
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Pareto Efficiency
• The strategy profile s Pareto dominates the strategy profile s’ if for some players the utility for s is strictly higher and for the others is not worse
• A strategy profile is Pareto optimal if there is no other strategy profile dominating it
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Nash Equilibrium• Player’s i best response to the strategy
profile s-i is a mixed strategy s*i ∈Si such that ui(s*i ,s-i)≥ui(si, s-i) for every si ∈Si
• A strategy profile s=(s1,...,sn) is a Nash equilibrium if, for all agents , si is a best response to s-i
• Weak Nash (≥), Strong Nash (>)
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Battle of the SexesLW WL
LW
WL
2,1 0,0
0,0 1,2
• Both pure strategies are Nash Equilibria
• Are there any other Nash Equilibria?
• There is at least another mixed-strategy equilibrium (usually very tricky to compute, but can be done with simple examples)
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Battle of the Sexes• Suppose the husband chooses LW
with probability p and WL with probability p-1
• The wife should be indifferent between her available options, otherwise she would be better off choosing a pure strategy
• What is the p which allows the wife to be really indifferent?
• Please notice that the pure strategies are Pareto optimal
LW WL
LW
WL
2,1 0,0
0,0 1,2
Being “indifferent” means obtaining the same utility, not
playing indifferently
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Battle of the Sexes
Uwife (LW) =Uwife (WL)1·p + 0·(1− p) = 0·p + 2·(1− p)
p = 2 − 2p p = 23
Uhusband (LW) =Uhusband (WL)2·r + 0·(1− r) = 0·r +1·(1− r)
2r = 1− r r = 13
p: probability that husband plays LW r: probability that wife plays LW
Uw (s) = 2(1− p)(1− r)+ pr
Uw sw (r),23LW+ 1
3WL⎛
⎝⎜⎞⎠⎟ =
23(1− r)+ 2
3r = 2
3Uh (s) = (1− p)(1− r)+ 2pr
Uh sh (p),13LW+ 2
3WL⎛
⎝⎜⎞⎠⎟ =
23(1− p)+ 2
3p = 2
3
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Matching PenniesHeads Tails
Heads
Tails
1,-1 -1,1
-1,1 1,-1
• Do we have any pure strategies?
• No
• Do we have mixed strategies?
• Yes
U1(H) =U1(T)1·p + (−1)·(1− p) = −1·p +1·(1− p)
2p −1= 1− 2p p = 12
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Rock-Paper-ScissorRock Paper Scissors
Rock
Paper
Scissors
0,0 -1,1 1,-1
1,-1 0,0 -1,1
-1,1 1,-1 0,0
• Do we have any pure strategies?
• No
• Do we have mixed strategies?
• Yespr + pp + ps = 1U1(R) =U1(P) =U1(S)
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Rock-Paper-ScissorR P S
RPS
0,0 -1,1 1,-11,-1 0,0 -1,1-1,1 1,-1 0,00pr + (−1)pp +1ps = 1pr + 0pp + (−1)ps
1pr + 0pp + (−1)ps = (−1)pr +1pp + 0ps
⎧⎨⎩
ps =13
pr =13
pp =13
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
ps = pp = pr
2ps = pr + pp2pr = ps + pp
⎧⎨⎩
2ps =ps + pp2
+ pp
4 ps = ps + 3pp
pr + pp + ps = 1U1(R) =U1(P) =U1(S)
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Existence of Nash Equilibria• We have seen that not every game has a
pure strategy Nash equilibrium
• Does every game have a Nash equilibrium (random or pure)?
Theorem (Nash, 1951) Every game with a finite number of players and action profiles has at least one Nash equilibrium
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Computing Nash Equilibria
• There are algorithms which compute Nash equilibria, but they are exponential in the size of the game
• It is not known if there are polynomial algorithms (but the consensus is that there are none)
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Dominated StrategiesDefinitions
• Let si and si’ be two strategies of player i and S-i the set of all strategy profiles of the remaining players
• si strictly dominates si’ if for all s-i ∈S-i, it is the case that ui(si,s-i)>ui(si’,s-i)
• si weakly dominates si’ if for all s-i ∈S-i, it is the case that ui(si,s-i)≥ui(si’,s-i) and for at least one s-i ∈S-i it is the case that ui(si,s-i)>ui(si’,s-i)
• si very weakly dominates si’ if for all s-i ∈S-i, it is the case that ui(si,s-i)≥ui(si’,s-i)
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Dominated StrategiesExample
L C R
U
M
D
3,1 0,1 0,0
1,1 1,1 5,0
0,1 4,1 0,0
L C
U
M
D
3,1 0,1
1,1 1,1
0,1 4,1
L C
U
D
3,1 0,1
0,1 4,1
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Dominated StrategiesPrisoner’s Dilemma
C D
C
D
-1,-1 -4,0
0,-4 -3,-3
C D
D 0,-4 -3,-3
D
D -3,-3
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Dominated Strategiesas solution concepts
• The set of all strategy profiles that assign 0 probability to playing any action that would be removed through iterated removal of strictly dominated strategies is a solution concept
• Sometimes, no action can be “removed”, sometimes we can solve the game (we say the game is solvable by iterated elimination)
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Dominated StrategiesCosts
• Iterated elimination ends after a finite number of steps
• Iterated elimination preserves Nash equilibria
• We can use it to reduce the size of the game
• Iterated elimination of strictly dominated strategies can occur in any order without changing the results
• Checking if a (possibly mixed) strategy is dominated can be done in polynomial time
• Domination by pure strategies can be checked with a very simple iterative algorithm
• Domination by mixed strategies can be checked solving a linear problem
• Iterative elimination needs only to check pure strategies
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Dominated Strategies(domination by pure-strategies)forall pure strategies ai∈Ai for player i where ai≠si dodom ← trueforall pure-‐strategy profiles a-‐i∈A-‐i doif ui(si,a-‐i)≥ui(ai,a-‐i) thendom ← falsebreak
if dom = true thenreturn true
return false
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• The maxmin strategy for player i in an n player, general sum game is a not necessarily unique (mixed) strategy that maximizes i’s worst case payoff
• The maxmin value (or security level) is the minimum payoff level guaranteed by a maxmin strategy
• In two player general sum games the minmax strategy for player i against player –i is the strategy that keeps the maximum payoff for –i at minimum
• It is a punishment
Other Solution Concepts:Maxmin & Minmax
argmaxsi mins− i ui (si , s−i )
argminsi maxs− i ui (si , s−i )
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Other Solution Concepts:Minmax, n-player
• In an n-player game, the minmax strategy for player i against player j ≠ i is i’s component of the mixed-strategy profile s-j in the expression:
where –j denotes the set of players other that j.
• Player i receives his minmax value if players –i choose their strategies to minimize i utility “after” he chose strategy si
• A player maxmin value is always less than or equal to his minmax value
argmins− j maxs j u j (s j , s− j )
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Maxmin & Minmax Examples• Matching Pennies
• Maxmin: 0.5 T + 0.5 H
• Minmax: 0.5 T + 0.5 H
• Battle of Sexes
• Maxmin: H→0.66 LW + 0.33 WL W→0.33 LW + 0.66 WL
• Minmax: H→0.66 LW + 0.33 WL W→0.33 LW + 0.66 WL
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Minmax Theorem
• Each player’s maxmin equals his minmax (value of the game)
• Maxmin strategies coincide with minmax strategies
• Any maxmin strategy profile is a Nash equilibrium and any Nash equilibrium is a maxmin strategy profile
Theorem (von Neumann, 1928) In any finite, two-player, zero-sum game, in any Nash equilibrium each player receives a payoff that is equal to both his maxmin value and his minmax value
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0
0.3 0.6
0.9
-1
-0.5
0
0.5
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
P2 P(heads)
P1
Exp
ecte
d U
tilit
y
P1 P(heads)
Matching Pennies for P1
Matching Pennies
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Minimax Regret• An agent i’s regret for playing an action ai if
other agents adopt action profile a-i is defined as:
• An agent i’s maximum regret for playing an action ai is defined as:
• Minimax regret actions for agent i are defined as:
maxa ′i ∈Ai
ui a ′i ,a−i( )⎡⎣⎢
⎤⎦⎥− ui ai ,a−i( )
maxa− i∈A− i
maxa ′i ∈Ai
ui a ′i ,a−i( )⎡⎣⎢
⎤⎦⎥− ui ai ,a−i( )⎛
⎝⎜⎞⎠⎟
argminai∈Ai
maxa− i∈A− i
maxa ′i ∈Ai
ui a ′i ,a−i( )⎡⎣⎢
⎤⎦⎥− ui ai ,a−i( )⎛
⎝⎜⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥
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Maxmin vs. Minmax RegretL R
T
B
100, a 1-!, b
2, c 1, d
P1 Maxmin is B (why?), his Minimax Regret strategy is T
regret(T ,[R])= 1−1+ = regret(B,[R]) = 1−1= 0regret(T ,[L]) = 100 −100 = 0regret(B,[L]) = 100 − 2 = 98
max regret(T )= max{,0} = maxregret(B) = max{98,0} = 98
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Computing Equilibria
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Computing Nash-Equilibrium for two-players zero-sum games
• Consider the class of two-player zero-sum games
• is the expected utility for player i in equilibrium
• In the next slide, the LP for computing player 2 and player 1 strategies are given
• Linear Programs are rather inexpensive to compute
Ui*
Γ = 1,2{ },A1 × A2 , u1,u2( )( )
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Computing Nash-Equilibrium for two-players zero-sum games
• Constants: u1(...)
• Variables: s2, U1*
minimize U1*
subject to u1 a1j ,a2
k( ) ⋅ s2k ≤U1
*
k∈A2
∑ ∀j ∈A1
s2k = 1
k∈A2
∑s2k ≥ 0 ∀k ∈A2
U1* is a maxmin value!
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Computing Nash-Equilibrium for two-players zero-sum games
• Constants: u1(...)
• Variables: s1, U1*
maximize U1*
subject to u1 a1j ,a2
k( ) ⋅ s1j ≥U1
*
j∈A1
∑ ∀k ∈A2
s1j = 1
j∈A1
∑s1j ≥ 0 ∀j ∈A1
U1* is a maxmin value!
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Computing Nash-Equilibrium for two-players zero-sum games
• Constants: u1(...)
• Variables: s2, U1*
minimize U1*
subject to u1 a1j ,a2
k( ) ⋅ s2k + r1
j =U1*
k∈A2
∑ ∀j ∈A1
s2k = 1
k∈A2
∑s2k ≥ 0 ∀k ∈A2
r1j ≥ 0 ∀j ∈A1
U1* is a maxmin value!
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Computing Nash-Equilibrium for two-players zero-sum games
• Constants: u1(...)
• Variables: s2, U1*
minimize U1*
subject to u1 a1j ,a2
k( ) ⋅ s1j + r2
k =U1*
j∈A1
∑ ∀k ∈A2
s1j = 1
j∈A1
∑s1j ≥ 0 ∀j ∈A1
r1k ≥ 0 ∀k ∈A2
U1* is a maxmin value!
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Computing maxmin & minmaxfor two-players general-sum games
• We know how to compute minmax & maxmin strategies for two-players 0-sum games
• It is sufficient to transform the general-sum game in a 0-sum game
• Let G be an arbitrary two-player game G=({1,2}, A1⨉A2,(u1,u2)); we define G’= ({1,2}, A1⨉A2,(u1,-u1))
• G’ is 0-sum: a strategy that is part of a Nash equilibrium for G’ is a maxmin strategy for 1 in G’
• Player 1 maxmin strategy is independent of u2
• Thus player’s 1 maxmin strategy is the same in G and in G’
• A minmax strategy for Player 2 in G’ is a minmax strategy for 2 in G as well (for the same reasons)
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Two Players General sum Games
u1 a1j ,a2
k( ) ⋅ s2k + r1j =U1*
k∈A2∑ ∀j ∈A1
u1 a1j ,a2
k( ) ⋅ s1j + r2k =U1*
j∈A1∑ ∀k ∈A2
s1j = 1
j∈A1∑ s2
k = 1k∈A2∑
s1j ≥ 0, s2
k ≥ 0 ∀j ∈A1,∀k ∈A2r1j ≥ 0, r2
k ≥ 0 ∀j ∈A1,∀k ∈A2r1j ·s1
j = 0, r2k ·s2
k = 0 ∀j ∈A1,∀k ∈A2
• We can formulate the game as a linear complimentarity problem (LCP)
• This is a constraint satisfaction problem (feasibility, not optimization)
• The Lemke-Howson algorithm is the best suited to solve this kind of problems
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Computing n-players Nash• Could be formulated as a nonlinear
complementarity problem (NLCP), thus it would not be easily solvable
• A sequence of linear complementarity problems (SLCP) can be used; it is not always convergent, but if we’re lucky it’s fast
• It is possible to formulate as the computation of the minimum of a specific function, both with or without constraints
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Software Tools
McKelvey, Richard D., McLennan, Andrew M., and Turocy, Theodore L. (2010). Gambit: Software Tools for Game Theory, Version 0.2010.09.01.http://www.gambit-project.org.
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Other Games
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Outline Introduction Game Representations Reductions Solution Concepts
Extended Form of Game-1
(red)
foldraise
meet
pass
meet
passraise
fold
(black)
.5
.5
(2,-2)
(-2,2)
(1,-1)
(1,-1)
(1,-1)
(-1,1)
0
1.a 2.0
2.01.b
Enrico Franchi Game Theory
Extensive Form Game Example
• Each player bets a coin
• Player 1 draw a card and is the only one to see it
• Player 1 also plays before Player 2
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Repeated Games
• What happens if the same NF game is repeated:
• An infinite number of times?
• A finite number of times?
• A finite but unknown number of times?
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Bayesian Games• Represent uncertainty about the game
being played; there is a set of possible games
• with the same number of agents and same strategy spaces but different payoffs
• Agents beliefs are posteriors obtained conditioning a common prior on individual private signals