Nash Equilibrium In NonZero

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Nash Equilibrium In Non-Zero Sum Games
CMSC828N Spring 2009
Emre Sefer
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Introduction

• Non-zero sum games are more complicated than
  zero-sum games since the interests of the players
  may not be mutually opposed.
• Neither may their interests be mutually
  coincident

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Nash equilibrium
Nash Equilibrium - Motivation We wish to find
plausible outcomes in games for example,
in order to predict the outcome What is a
plausible outcome? Possible criteria No
player is surprised by the outcome No
player acts against her best interest given her
expectations Action profile
would be repeated if the same people
play again, i.e. nobody would want to change his
action Profile forms a steady state if we
randomly rematch players drawn from
large populations
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Nash Equilibrium

• In a Nash equilibrium each player chooses
  according to rational choice given her/his
  beliefs about other players actions and all
  players beliefs are correct
• NE is such set of strategies, that no player is
  willing to deviate to any other pure strategy.
• Equilibrium should be rational, optimal and
  stable In equilibrium player has no incentive to
  change behavior because payoff cannot be
  increased by doing so
• In other words, the strategies are multilateral
  best responses

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Nash equilibrium-formal definition
Nash Equilibrium Formal Definition Let Ai be
the set of actions available for player i a
(a1, a2, , ai,) be an action profile write
(ai', a-i) if i chooses ai', other
players according to a Then a is a Nash
equilibrium (of a strategic game with ordinal
preferences) if for every player i and every
action all ai that is element of Ai
ui(a) ui (ai , a-i) where ui is the payoff
function representing the preferences of player
i This means given all players follow a,
no individual player would want to deviate they
could, however, be jointly better off
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Examples
Example 1 The Prisoners Dilemma The
unique Nash equilibrium is (D,D) For every
other profile, at least one player wants to
deviate It is actually irrelevant here what
players believe, they prefer D anyway.
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Examples
Example 2 Battle of Sexes There are two
Nash equilibria (Ball, Ball) and (Theatre,
Theatre) Question is Which one to
choose?
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Focal points

• For the games with multiple Nash Equilibria
• In some of these games, some of these equilibria
  seem more likely to attract the players
  attentions than others, i.e. some equilibria are
  focal(Schelling, 1960)

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Examples
Example 3 Variant of Battle of
Sexes There are again two Nash
equilibria (Ball, Ball) and (Theatre, Theatre)
But now the choice seems easy Ball equilibrium
is focal point (by Schelling)
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Examples

• Example 4 Stag Hunt
• There are two equilibria(Stag, Stag) and (Hare,
  Hare)
• Which equilibrium to choose? Stag could be
  focal
• If there are more players, there are still only
  two equilibria
• either all choose Stag or all choose Hare

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Strict Nash Equilibrium
In previous examples, each player was always
strictly better off than if player deviated
This is not required by Nash equilibrium
Example? If it holds then we talk of a strict
Nash equilibrium An action profile a is a
strict Nash equilibrium (of a strategic game with
ordinal preferences) if for every player i and
every action ai that is element of Ai
ui(a) gt ui (ai, a-i) where ui is the
payoff function representing the preferences of
player i
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Strictly Dominated Strategies
Finding a Nash equilibrium is sometimes
easy because we can exclude dominated
strategies ai' strictly dominates ai if for all
strategy profiles a-i of the other
players ui(ai', a-i) gt ui (ai, a-i) ai is then
strictly dominated a strictly dominated
strategy cannot be chosen in any Nash equilibrium
(why?) Thus we can start by eliminating
strictly dominated strategies Example
Prisoners dilemma, C is strictly dominated
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Dominance example
B is dominated for Player 1 After eliminating
B,R is dominated for Player 2
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Dominant Strategies
If a strategy strictly dominates all other
strategies, it is strictly dominant ai' is a
strictly dominant strategy for player i if for
all ai?ai' and all strategy profiles a-i of the
other players ui(ai', a-i) gtui(ai, a-i) If a
player has a strictly dominant strategy, this
must be played in a Nash equilibrium (why?)
Then we can fix ai' and continue by finding the
other players best responses
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Strictly Dominant Example

• L is strictly dominant.
• Nash Equilibrium must be in L

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Iterated Elimination of Dominated Strategies
When searching for Nash equilibria, we can
eliminate strictly dominated strategies What
happens then? If after eliminating strictly
dominated strategies a player has again a
strictly dominated strategy, this cannot be
played in any Nash equilibrium (why?) All Nash
equilibria must be among the surviving profiles,
but not all surviving profiles are Nash-
equilibria
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Iterated Elimination of Dominated Strategies

• Iterated elimination of dominated strategies
  (IEDS) provides a solution to many normal-form
  games
• Step 1 Identify all dominated strategies
• Step 2 Eliminate them to obtain a reduced game
• Step 3 Go to Step 1

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IEDS Example
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IEDS Example
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IEDS Example
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IEDS Example
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Another IEDS Example
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Weakly Dominated Strategies
ai' weakly dominates ai if for all strategy
profiles a-i of the other players ui(ai', a-i)
ui(ai, a-i) and there is at least one a-i'
such that ui(ai', a-i') gtui(ai, a-i') ai is
then weakly dominated a weakly dominated
strategy can be chosen in a Nash equilibrium
While (iterated) elimination of weakly dominated
strategies is plausible and sometimes leads to a
Nash equilibrium, it is problematic because we
may eliminate equilibria and the order of
elimination matters
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Order of Elimination

• Question Does the order of elimination
  matter?
• Answer Although it is not obvious, the end
  result of iterated strict dominance is always the
  same regardless of the sequence of eliminations.
• If in some game you can either eliminate U for
  Player 1 or L for Player 2, you dont need to
  worry about which one to do first either way
  youll end up at the same answer.
• This result only holds under iterated strict
  dominance according to which we eliminate a
  strategy only if there is some other
• strategy that yields payoffs that are strictly
  higher no matter what the other players do.

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Order of Elimination

• If you eliminate a strategy when there is some
  other strategy that yields payoffs that are
  higher or equal no matter what the other players
  do, you are doing iterated weak dominance
• In this case you will not always get the same
  answer regardless of the sequence of
  eliminations.
• This is a serious problem, and this is the reason
  why iterated strict dominance is mostly used.

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Order of Elimination

• In the example below
• U is weakly dominated by M for Player1
• M is weakly dominated by D for Player 1

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Order of Elimination

• First solution
• Eliminate U and then M
• NE are (D,L) and (D,R)
• Second solution
• Eliminate U and then R
• NE is (D,L)

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Best Response Functions
We could find the equilibrium just by going
through the few possible action profiles. This
is getting messy if there are many
available actions (in particular if there are
infinitely many) Consider best
response function of i Bi(a-i) ai is
element of Ai ui(ai,a-i ) ui(ai',a-i ) for all
ai that is element of Ai Set-valued, each
member of Bi(a-i) is a best response to a-i
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Best Response Functions
Proposition a is a Nash equilibrium if and only
if ai is element of Bi(a-i)
for every i (1) If all players have
unique best responses for each combination of
others actions, i.e. Bi(a-i)
bi(a-i) then (1) becomes ai
bi(a-i) for every i (2) To find Nash
equilibrium find best response functions for
each player find a that satisfies (1) (or (2)
if best responses have only one value)
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Best Response Function Examples
Two players divide 10. Players make demands a1,
a2. If a1 a2 10, they get a1, a2. If a1 a2 gt
10 and ai 5, i gets ai and j gets 10 ai. If
a1 gt 5 and a2 gt 5, they both get 5.
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Best Response Function Examples
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Best Response Function Examples
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Best Response Function Examples

• What is Player 1s best response to Player 2s
• strategy of B1, B2 or B3?
• What is Player 2s best response to Player 1s
• strategy of A1, A2 or A3?

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Best Response

• M is Player 1s best response to Player 2s
  strategy L
• T is Player 1s best response to Player 2s
  strategy C
• B is Player 1s best response to Player 2s
  strategy R
• L is Player 2s best response to Player 1s
  strategy T
• C is Player 2s best response to Player 1s
  strategy M
• R is Player 2s best response to Player 1s
  strategy B

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Best Response in 2-player game

• Using best response function to find Nash
  equilibrium in a 2-player game
• ( s1,s2) is a Nash equilibrium if and only if
• player 1s strategy s1 is her best response to
  player 2s strategy s2
• player 2s strategy s2 is her best response to
  player 1s strategy s1

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Battle of Sexes

• Ball is Player 1s best response to Player 2s
  strategy Ball
• Ball is Player 2s best response to Player 1s
  strategy Ball
• Hence, (Ball, Ball) is a Nash equilibrium
• Theatre is Player 1s best response to Player 2s
  strategy Theatre
• Theatre is Player 2s best response to Player 1s
  strategy Theatre
• Hence, (Theatre, Theatre) is a Nash equilibrium

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Matching Pennies

• Head is Player 1s best response to Player 2s
  strategy Tail
• Tail is Player 2s best response to Player 1s
  strategy Tail
• Tail is Player 1s best response to Player 2s
  strategy Head
• Head is Player 2s best response to Player 1s
  strategy Head
• Hence, NO Nash equilibrium

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Symmetric Games
Definition (Symmetric Two-Player Game) A
two-player game with ordinal preferences is
symmetric if the players sets of actions are the
same and the players preferences are represented
by payoff functions u1 and u2 for which
u1(a1,a2) u2(a2,a1) for every action pair
(a1,a2) Examples Prisoners Dilemma is
symmetric Battle of the Sexes is not symmetric
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Symmetric NE

• Definition (Symmetric NE) An action profile a in
  a strategic game with ordinal preferences in
  which each player has the same set of actions is
  ac symmetric Nash Equilibrium if it is
• a Nash Equilibrium
• Ai is the same for every player I
• Example

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References
1-The Strategy of Conflict (1960), Thomas
Schelling 2-Royal Holloway University of London
EC 3224 course lecture notes 3-Game Theory.
Fudenberg and Tirole 4-A course in Game Theory,
Osborne and Rubinstein
THANK YOU ! Questions?