Final Exam
Stat155
Game Theory
Lecture 27: Final Review
“This is an open book exam: you can use any printed or written material,
but you cannot use a laptop, tablet, or phone (or any device that can
communicate). Answer each question in the space provided.”
Peter Bartlett
December 6, 2016
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Topics 1
Topics 2
Combinatorial games
Positions, moves, terminal positions, impartial/partisan, progressively
bounded
Progressively bounded impartial and partisan games
The sets N and P
Theorem: Someone can win
Examples: Subtraction, Chomp, Nim, Rims, Staircase Nim, Hex
General-sum games, Nash equilibria.
Two-player: payoff matrices, dominant strategies, safety strategies.
Multiplayer: Utility functions, Nash’s Theorem
Congestion games and potential games
Every potential game has a pure Nash equilibrium
Every congestion game is a potential game
Zero sum games
Other equilibrium concepts
Payoff matrices, pure and mixed strategies, safety strategies
Von Neumann’s minimax theorem
Solving two player zero-sum games
Evolutionarily stable strategies
Correlated equilibrium
Price of anarchy
Saddle points
Equalizing strategies
Solving 2 × 2 games
Dominated strategies
2 × n and m × 2 games
Principle of indifference
Symmetry: Invariance under permutations
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
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Topics 3
Definitions: Combinatorial games
Cooperative games
Transferable versus nontransferable utility
Two-player transferable utility cooperative games
A combinatorial game has:
Cooperative strategy, threat strategies, disagreement point, final payoff
vector
Two-player nontransferable utility cooperative games
Two players, Player I and Player II.
A set of positions X .
For each player, a set of legal moves between positions, that is, a set
of ordered pairs, (current position, next position):
Bargaining problems
Nash’s bargaining axioms, the Nash bargaining solution
Multi-player transferable utility cooperative games
MI , MII ⊂ X × X .
Characteristic function
Gillies’ core
Shapley’s axioms, Shapley’s Theorem
Players alternately choose moves.
Designing games
Voting systems.
Play continues until some player cannot move.
Voting rules, ranking rules
Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem
Properties of voting rules
Instant runoff voting, Borda count, positional voting rules
Normal play: the player that cannot move loses the game.
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Definitions: Combinatorial games
Impartial combinatorial games and winning strategies
Terminology:
An impartial game has the same set of legal moves for both players:
MI = MII .
A partisan game has different sets of legal moves for the players.
A terminal position for a player has no legal move to another position.
x is terminal for player I if there is no y ∈ X with (x, y ) ∈ MI .
A combinatorial game is progressively bounded if, for every starting
position x0 ∈ X , there is finite bound on the number of moves before
the game ends. (That is, if B(x) denotes the maximum number of
moves before the game ends, then B(x) < ∞.)
A winning strategy for a player from position x:
a mapping from non-terminal positions to legal moves that is
guaranteed to result in a win for that player from that position.
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Theorem
In a progressively bounded impartial
combinatorial game under normal
play, X = N ∪ P.
That is, from any initial position, one
of the players has a winning strategy.
Proof: induction on number of
moves until the end.
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Key Ideas: Progressively bounded impartial games
Example: Nim
k piles of chips.
Remove some (positive) number of chips from some pile.
A player wins when they take the last chip.
Bouton’s Theorem
P: Every move leads to N.
A Nim position (x1 , . . . , xk ) is in P iff
the Nim-sum of its components is 0.
N: Some move leads to P (hence cannot contain terminal positions).
The Nim-sum of x = (x1 , . . . , xk ) is written x1 ⊕ x2 ⊕ · · · ⊕ xk .
The binary representation of the Nim-sum is the bitwise sum, in
modulo-two arithmetic, of the binary representations of the
components of x.
Proof: Show that Z := {(x1 , . . . , xk ) : x1 ⊕ · · · ⊕ xk = 0} is P:
Z must lead to Z c ; from Z c , there is a move into Z .
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Partisan Games
Topics 1
Combinatorial games
Recall:
An impartial game has the same set of legal moves for both players:
MI = MII .
A partisan game has different sets of legal moves for the players.
Positions, moves, terminal positions, impartial/partisan, progressively
bounded
Progressively bounded impartial and partisan games
The sets N and P
Theorem: Someone can win
Examples: Subtraction, Chomp, Nim, Rims, Staircase Nim, Hex
Zero sum games
Theorem
Payoff matrices, pure and mixed strategies, safety strategies
Von Neumann’s minimax theorem
Solving two player zero-sum games
Consider a progressively bounded partisan
combinatorial game under normal play, with no ties
allowed. From any initial position, one of the players
has a winning strategy.
Saddle points
Equalizing strategies
Solving 2 × 2 games
Dominated strategies
2 × n and m × 2 games
Principle of indifference
Symmetry: Invariance under permutations
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Two-player zero-sum games
Two-player zero-sum games
Definitions
Player I has m actions, 1, 2, . . . , m.
Definitions
Player II has n actions, 1, 2, . . . , n.
A mixed strategy is a probability distribution over actions.
The payoff matrix A ∈ Rm×n represents the payoff to Player I:


a11 a12 · · · a1n
 a21 a22 · · · a2n 


A= .
.. 
..
 ..
.
. 
am1 am2 · · ·
amn
A mixed strategy for Player I is a vector
 
x1
(
)
m
 x2 
X
 
x =  .  ∈ ∆m := x ∈ Rm : xi ≥ 0,
xi = 1 .
 .. 
i=1
xm
If Player I chooses i and Player II chooses j, the payoff to Player I is
aij and the payoff to Player II is −aij .
The sum of the payoff to Player I and the payoff to Player II is 0.
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Two-player zero-sum games
Two-player zero-sum games
The expected payoff to Player I when Player I plays mixed strategy
x ∈ ∆m and Player II plays mixed strategy y ∈ ∆n is
A mixed strategy for Player II is a vector
 
y1
(
)
n
y2 
X
 
yi = 1 .
y =  .  ∈ ∆n := y ∈ Rn : yi ≥ 0,
 .. 
yn
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EI ∼x EJ∼y aIJ =
m X
n
X
xi aij yj
i=1 j=1
= x > Ay
i=1
= x1 x2 · · ·
A pure strategy is a mixed strategy where one entry is 1 and the
others 0. (This is a canonical basis vector ei .)

a11

 a21
xm  .
 ..
a12
a22
···
···
..
.
am1 am2 · · ·
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 
a1n
y1
 y2 
a2n 
 
..   .. 
.  . 
amn
yn
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Two-player zero-sum games
Two-player zero-sum games
Von Neumann’s Minimax Theorem
A safety strategy for Player I is an
x∗
∈ ∆m that satisfies
For any two-person zero-sum game with payoff matrix A ∈ Rm×n ,
min x ∗ > Ay = max min x > Ay .
y ∈∆n
max min x > Ay = min max x > Ay .
x∈∆m y ∈∆n
x∈∆m y ∈∆n
This mixed strategy maximizes the worst case expected gain for
Player I.
We call the optimal expected payoff the value of the game,
V := max min x > Ay = min max x > Ay .
Similarly, a safety strategy for Player II is a y ∗ ∈ ∆n that satisfies
x∈∆m y ∈∆n
max x > Ay ∗ = min max x > Ay .
y ∈∆n x∈∆m
LHS: Player I plays x ∈ ∆m first, then Player II responds with y ∈ ∆n .
RHS: Player II plays y ∈ ∆n first, then Player I responds with x ∈ ∆m .
Notice that we should always prefer to play last:
y ∈∆n x∈∆m
x∈∆m
y ∈∆n x∈∆m
This mixed strategy minimizes the worst case expected loss for
Player II.
max min x > Ay ≤ min max x > Ay .
x∈∆m y ∈∆n
y ∈∆n x∈∆m
The astonishing part is that it does not help.
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Two-player zero-sum games
Topics 1
Combinatorial games
Von Neumann’s Minimax Theorem
Positions, moves, terminal positions, impartial/partisan, progressively
bounded
Progressively bounded impartial and partisan games
For any two-person zero-sum game with payoff matrix A ∈ Rm×n ,
max min x > Ay = min max x > Ay .
x∈∆m y ∈∆n
Zero sum games
y ∈∆n x∈∆m
Payoff matrices, pure and mixed strategies, safety strategies
Von Neumann’s minimax theorem
Solving two player zero-sum games
Safety strategies are optimal strategies:
For safety strategies x ∗ for Player I and y ∗ for Player II,
Saddle points
Equalizing strategies
Solving 2 × 2 games
Dominated strategies
2 × n and m × 2 games
Principle of indifference
Symmetry: Invariance under permutations
min x ∗ > Ay = max x > Ay ∗ = x ∗ > Ay ∗ = V .
y ∈∆n
x∈∆m
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Saddle points
Saddle points
Definition
A pair (i ∗ , j ∗ ) ∈ {1, . . . , m} × {1, . . . , n} is a saddle point for a payoff
matrix A ∈ Rm×n if
Theorem
If (i ∗ , j ∗ ) is a saddle point for a payoff matrix A ∈ Rm×n , then
max aij ∗ = ai ∗ j ∗ = min ai ∗ j .
i
ei ∗ is an optimal strategy for Player I,
j
ej ∗ is an optimal strategy for Player II, and
i∗
If Player I plays and Player II plays
incentive to change.
j ∗,
neither player has an
the value of the game is ai ∗ j ∗ .
Think of these as locally optimal strategies for the players.
They are also globally optimal strategies.
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2 × 2 games
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Dominated pure strategies
How to solve a 2 × 2 game
1
Check for a saddle point.
(Is the max of row mins = min of column maxes?)
2
If there are no saddle points, find equalizing strategies.
Definition
A pure strategy ei for Player I is dominated by ei 0 in payoff matrix A if, for
all j ∈ {1, . . . , n},
aij ≤ ai 0 j .
Equalizing strategies satisfy:
x1 a11 + (1 − x1 )a21 = x1 a12 + (1 − x1 )a22 ,
y1 a11 + (1 − y1 )a12 = y1 a21 + (1 − y1 )a22 .
Solving gives
a21 − a22
,
a21 − a22 + a12 − a11
a12 − a22
y1 =
.
a12 − a22 + a21 − a11
x1 =
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Solving 2 × n games
Principle of indifference
Theorem
Payoff matrix
2 3 1 5
4 1 6 0
Suppose a game with payoff matrix A ∈ Rm×n has value V . If x ∈ ∆m
and y ∈ ∆n are optimal strategies for Players I and II, then
for all j,
The maximum occurs at the
intersection of the lines
corresponding to columns 2 and 3.
if yj > 0,
(Ferguson, 2014)
The optimal strategy for Player II
involves only Columns 2 and 3. We
can find it by solving a 2 × 2 game.
l=1
m
X
xl alj ≥ V ,
for all i,
xl alj = V ,
if xi > 0,
l=1
The optimal strategy for Player I is
x = (5/7, 2/7).
Then Player II is indifferent between
columns 2 and 3.
m
X
n
X
l=1
n
X
ail yl ≤ V ,
ail yl = V .
l=1
This means that if one player is playing optimally, any action that has
positive weight in the other player’s optimal mixed strategy is a
suitable response.
It implies that any mixture of these “active actions” is also a suitable
response.
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Using the principle of indifference
Example
Diagonal payoff matrix
Solving linear systems
Suppose that we have a payoff matrix A and we suspect that an
optimal strategy for Player I has certain components positive, say
x1 > 0, x3 > 0.
Then we can solve the corresponding “indifference equalities” to find
y , say
n
X
l=1
a1l yl = V ,
n
X


a11 0
0
A =  0 a22 0 
0
0 a33
The aii are all positive, so we suspect that all xi , yi > 0 for the
optimal strategies.
Solve
a3l yl = V .
>
x A= V
l=1
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V
V :
x =y =
V =
V
a11
V
a22
V
a33
>
,
1
.
1/a11 + 1/a22 + 1/a33
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Topics 1
Symmetry
Combinatorial games
Positions, moves, terminal positions, impartial/partisan, progressively
bounded
Progressively bounded impartial and partisan games
Zero sum games
Payoff matrices, pure and mixed strategies, safety strategies
Von Neumann’s minimax theorem
Solving two player zero-sum games
Definition
A game with payoff matrix A ∈ Rm×n is invariant under a permutation πx
on {1, . . . , m} if there is a permutation πy on {1, . . . , n} such that, for all
i, j, aij = aπx (i),πy (j) .
If A is invariant under permutations π1 and π2 on {1, . . . , m}, then it
is invariant under π1 ◦ π2 .
Saddle points
Equalizing strategies
Solving 2 × 2 games
Dominated strategies
2 × n and m × 2 games
Principle of indifference
Symmetry: Invariance under permutations
If A is invariant under some set S of permutations, then it is invariant
under the group G of permutations generated by S (that is,
compositions and inverses).
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Symmetry
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Topics 1
Combinatorial games
Definition
Positions, moves, terminal positions, impartial/partisan, progressively
bounded
Progressively bounded impartial and partisan games
A mixed strategy x ∈ ∆m is invariant under a permutation πx on
{1, . . . , m} if for all i, xi = xπx (i) .
Zero sum games
An orbit of a group G of permutations is a set Oi = {π(i) : π ∈ G }.
If a mixed strategy x is invariant under a group G of permutations,
then for every orbit, x is constant on the orbit.
Theorem
If A is invariant under a group G of permutations, then there are optimal
strategies that are invariant under G .
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Payoff matrices, pure and mixed strategies, safety strategies
Von Neumann’s minimax theorem
Solving two player zero-sum games
Saddle points
Equalizing strategies
Solving 2 × 2 games
Dominated strategies
2 × n and m × 2 games
Principle of indifference
Symmetry: Invariance under permutations
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Topics 2
Noncooperative games
General-sum games, Nash equilibria.
Two-player: payoff matrices, dominant strategies, safety strategies.
Multiplayer: Utility functions, Nash’s Theorem
Players play their strategies simultaneously.
Congestion games and potential games
Every potential game has a pure Nash equilibrium
Every congestion game is a potential game
They might communicate (or see a common signal; e.g., a traffic
signal), but there’s no enforced agreement.
Other equilibrium concepts
Natural solution concepts:
Nash equilibrium, correlated equilibrium.
No improvement from unilaterally deviating.
Evolutionarily stable strategies
Correlated equilibrium
Price of anarchy
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
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General-sum games
General-sum games
Dominated pure strategies
Notation
A two-person general-sum game is specified by two payoff matrices,
A, B ∈ Rm×n .
Simultaneously, Player I chooses i ∈ {1, . . . , m} and the Player II
chooses j ∈ {1, . . . , n}.
A pure strategy ei for Player I is dominated by ei 0 in payoff matrix A if, for
all j ∈ {1, . . . , n},
aij ≤ ai 0 j .
Similarly, a pure strategy ej for Player II is dominated by ej 0 in payoff
matrix B if, for all i ∈ {1, . . . , m},
Player I receives payoff aij .
Player II receives payoff bij .
bij ≤ bij 0 .
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General-sum games
General-sum games
Safety strategies
Nash equilibria
A safety strategy for Player I is an x∗ ∈ ∆m that satisfies
A pair (x∗ , y∗ ) ∈ ∆m × ∆n is a Nash equilibrium for payoff matrices
A, B ∈ Rm×n if
min x∗> Ay = max min x > Ay .
y ∈∆n
x∈∆m y ∈∆n
max x > Ay∗ = x∗> Ay∗ ,
x∈∆m
x∗ maximizes the worst case expected gain for Player I.
max x∗> By = x∗> By∗ .
Similarly, a safety strategy for Player II is a y∗ ∈ ∆n that satisfies
y ∈∆n
min x > By∗ = max min x > By .
x∈∆m
If Player I plays x∗ and Player II plays y∗ , neither player has an
incentive to unilaterally deviate.
y ∈∆n x∈∆m
y∗ maximizes the worst case expected gain for Player II.
x∗ is a best response to y∗ , y∗ is a best response to x∗ .
Although safety strategies are optimal for zero-sum games, they are
typically not optimal for general-sum games: they are too
conservative.
In general-sum games, there might be many Nash equilibria, with
different payoff vectors.
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Comparing two-player general-sum and zero-sum games
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Comparing two-player general-sum and zero-sum games
Zero-sum games
1
A pair of safety strategies is a Nash equilibrium (minimax theorem)
2
Hence, there is always a Nash equilibrium.
3
If there are multiple Nash equilibria, they form a convex set, and the
expected payoff is identical within that set.
Thus, any two Nash equilibria give the same payoff.
Zero-sum games
4
If each player has an equalizing mixed strategy
(that is, x > A = v 1> and Ay = v 1),
then this pair of strategies is a Nash equilibrium.
(from the principle of indifference)
General-sum games
General-sum games
4
1
A pair of safety strategies might be unstable.
(Opponent aims to maximize their payoff, not minimize mine.)
2
There is always a Nash equilibrium (Nash’s Theorem).
3
There can be multiple Nash equilibria, with different payoff vectors.
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If each player has an equalizing mixed strategy
for their opponent’s payoff matrix
(that is, x > B = v2 1> and Ay = v1 1),
then this pair of strategies is a Nash equilibrium.
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Topics 2
Multiplayer general-sum games
Notation
General-sum games, Nash equilibria.
A k-person general-sum game is specified by k utility functions,
uj : S1 × S2 × · · · × Sk → R.
Two-player: payoff matrices, dominant strategies, safety strategies.
Multiplayer: Utility functions, Nash’s Theorem
Player j can choose strategies sj ∈ Sj .
Congestion games and potential games
Every potential game has a pure Nash equilibrium
Every congestion game is a potential game
Simultaneously, each player chooses a strategy.
Player j receives payoff uj (s1 , . . . , sk ).
Other equilibrium concepts
Evolutionarily stable strategies
Correlated equilibrium
k = 2: u1 (i, j) = aij , u2 (i, j) = bij .
For s = (s1 , . . . , sk ), let s−i denote the strategies without the ith one:
Price of anarchy
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
s−i = (s1 , . . . , si−1 , si+1 , . . . , sk ).
And write (si , s−i ) as the full vector.
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Multiplayer general-sum games
Multiplayer general-sum games
Definition
Definition
A vector (s1∗ , . . . , sk∗ ) ∈ S1 × · · · × Sk is a pure Nash equilibrium for utility
functions u1 , . . . , uk if, for each player j ∈ {1, . . . , k},
max uj (sj , s∗−j ) = uj (sj∗ , s∗−j ).
sj ∈Sj
If the players play these sj∗ , nobody has an incentive to unilaterally
deviate: each player’s strategy is a best response to the other players’
strategies.
A sequence (x1∗ , . . . , xk∗ ) ∈ ∆S1 × · · · × ∆Sk (called a strategy profile) is a
Nash equilibrium for utility functions u1 , . . . , uk if, for each player
j ∈ {1, . . . , k},
max uj (xj , x∗−j ) = uj (xj∗ , x∗−j ).
xj ∈∆Sj
Here, we define
uj (x) = Es1 ∼x1 ,...,sk ∼xk uj (s1 , . . . , sk )
X
=
x1 (s1 ) · · · xk (sk )uj (s1 , . . . , sk ).
s1 ∈S1 ,...,sk ∈Sk
If the players play these mixed strategies xj∗ , nobody has an incentive
to unilaterally deviate: each player’s mixed strategy is a best response
to the other players’ mixed strategies.
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Multiplayer general-sum games
Multiplayer general-sum games
Nash’s Theorem (1951)
Every finite general-sum game has a Nash equilibrium.
Theorem
Proof Idea (two players)
Consider a strategy profile x ∈ ∆S1 × · · · × ∆Sk . Let
Ti = {s ∈ Si : xi (s) > 0}. The following statements are equivalent.
1
2
We find an “improvement” map M(x, y ) = (x̂, ŷ ), so that
x is a Nash equilibrium.
For each i, there is a ci such that
1
2
For all si ∈ Si , ui (si , x−i ) ≤ ci .
For si ∈ Ti , ui (si , x−i ) = ci .
←− No better response outside Ti
←− Indifferent within Ti
Compare with the principle of indifference in the zero-sum case.
1
x̂ > Ay > x > Ay (or x̂ > Ay = x > Ay , if x was a best response to y ),
2
x > B ŷ > x > By (or x > B ŷ = x > By , if y was a best response to x),
3
M is continuous.
It’s easy to find a map like this.
For instance, take a step in the direction of increasing payoff:
x̂ = P∆m (x + ηAy ),
ŷ = P∆n (y + ηB > x).
A Nash equilibrium is a fixed point of M.
The existence of a Nash equilibrium follows from Brouwer’s
fixed-point theorem.
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Nash’s Theorem
Topics 2
General-sum games, Nash equilibria.
Two-player: payoff matrices, dominant strategies, safety strategies.
Multiplayer: Utility functions, Nash’s Theorem
Congestion games and potential games
Every potential game has a pure Nash equilibrium
Every congestion game is a potential game
Brouwer’s Fixed-Point Theorem
A continuous map f : K → K from a convex, closed, bounded K ⊆ Rd
has a fixed point, that is, an x ∈ K satisfying f (x) = x.
Other equilibrium concepts
Evolutionarily stable strategies
Correlated equilibrium
Price of anarchy
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
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Congestion games: games with a pure Nash equilibrium
Congestion games
Definition
A congestion game has k players, m facilities {1, . . . , m} (e.g., edges)
Example
For player i, there is a set Si of strategies that are sets of facilities,
s ⊆ {1, . . . , m} (e.g., paths)
A
(1,2,4)
I, II, III
For facility j, there is a cost vector cj ∈ Rk , where cj (n) is the cost of
facility j when it is used by n players.
(2,3,5)
S
(2,3,5)
For a sequence s = (s1 , . . . , sn ), the utilities of the players are defined by
X
costi (s) = −ui (s) =
cj (nj (s)),
T
(1,2,6)
(2,4,8)
j∈si
B
where nj (s) = |{i : j ∈ si }| is the number of players using facility j.
Egalitarian: The utilities depend on how many players use each facility,
and not on which players use it.
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Congestion games
Congestion games
Proof
We define a potential function Φ : S1 × · · · × Sk → R as
Φ(s) =
j (s)
m nX
X
cj (l).
j=1 l=1
Theorem
Fix strategies for the k players s = (s1 , . . . , sk ).
What happens when Player i changes from si to si0 ?
∆costi = costi (si0 , s−i ) − costi (si , s−i )
X
X
=
cj (nj (s) + 1) −
cj (nj (s))
Every congestion game has a pure Nash equilibrium.
j∈(si0 −si )
|
{z
increased cost
}
= Φ(si0 , s−i ) − Φ(si , s−i )
j∈(si −si0 )
|
{z
decreased cost
}
= ∆Φ.
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The potential Φ reflects how a player’s costs change.
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Congestion games
Potential games
Proof
Definition
If we start at an arbitrary s, and update one player’s choice to
decrease that player’s cost, the potential must decrease.
Consider a multiplayer game G :
k players
Continuing updating other player’s strategies in this way, we must
eventually reach a local minimum (there are only finitely many
strategies).
For player i, there is a set Si of strategies.
Since no player can reduce their cost from there, we have reached a
pure Nash equilibrium.
This gives an algorithm for finding a pure Nash equilibrium: Update
the choice of one player at a time to improve their cost.
For player i, there is costi : S1 × · · · × Sk → R.
We say Φ : S1 × · · · × Sk → R is a potential for game G if ∆Φ = ∆costi ,
that is, for all i, s and si0 ,
Φ(si0 , s−i ) − Φ(si , s−i ) = costi (si0 , s−i ) − costi (si , s−i ).
We say that G is a potential game if it has a potential.
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Potential games
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Topics 2
General-sum games, Nash equilibria.
Two-player: payoff matrices, dominant strategies, safety strategies.
Multiplayer: Utility functions, Nash’s Theorem
Congestion games and potential games
In considering congestion games, we proved two things:
Every potential game has a pure Nash equilibrium
Every congestion game is a potential game
Theorem
1
Every congestion game is a potential game.
2
Every potential game has a pure Nash equilibrium.
Other equilibrium concepts
Evolutionarily stable strategies
Correlated equilibrium
Price of anarchy
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
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Alternatives to Nash equilibria?
Evolutionarily stable strategies
There is a population of individuals.
There is a game played between pairs of individuals.
Alternative equilibrium concepts
Correlated equilibrium
Each individual has a pure strategy encoded in its genes.
Evolutionary stability
The two players are randomly chosen individuals.
A higher payoff gives higher reproductive success.
This can push the population towards stable mixed strategies.
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Evolutionarily stable strategies
Evolutionarily stable strategies
Suppose that x is invaded by a small population of mutants z:
x is replaced by (1 − )x + z.
Will the mix x survive?
Consider a two-player game with payoff matrices A, B.
Suppose that it is symmetric (A = B > ).
x’s utility:
Consider a mixed strategy x.
z’s utility:
Think of x as the proportion of each pure strategy in the population.
x> A (z + (1 − )x) = x> Az + (1 − )x> Ax
z> A (z + (1 − )x) = z> Az + (1 − )z> Ax
Definition
Mixed strategy x ∈ ∆n is an evolutionarily stable strategy (ESS) if, for any
pure strategy z,
1
2
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z> Ax ≤ x> Ax
If z> Ax = x> Ax then z> Az < x> Az.
←− (x, x) is a Nash equilibrium.
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ESS and Nash equilibria
Topics 2
Theorem
General-sum games, Nash equilibria.
Two-player: payoff matrices, dominant strategies, safety strategies.
Multiplayer: Utility functions, Nash’s Theorem
Every ESS is a Nash equilibrium.
Definition
Congestion games and potential games
A strategy profile x = (x1∗ , . . . , xk∗ ) ∈ ∆S1 × · · · × ∆Sk is a strict Nash
equilibrium for utility functions u1 , . . . , uk if, for each player
j ∈ {1, . . . , k}, for all xj ∈ ∆Sj that is different from xj∗ ,
Other equilibrium concepts
Every potential game has a pure Nash equilibrium
Every congestion game is a potential game
Evolutionarily stable strategies
Correlated equilibrium
uj (xj , x∗−j ) < uj (xj∗ , x∗−j ).
Price of anarchy
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
Theorem
Every strict Nash equilibrium is an ESS.
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Correlated equilibria: A driving example
Correlated equilibrium
Payoff
Definition
Go
Stop
Go
(-100,-100)
(-1,1)
Stop
(1,-1)
(-1,-1)
For a two player game with strategy sets S1 = {1, . . . , m} and
S2 = {1, . . . , n}, a correlated strategy pair is a pair of random variables
(R, C ) with some joint probability distribution over pairs of actions
(i, j) ∈ S1 × S2 .
Nash equilibria
2
99
2
99
(Go, Stop), (Stop, Go), 101
, 101
, 101
, 101
.
(because we want indifference: −100p + 1 − p = −p − (1 − p))
Example
In the traffic signal example, Pr(Stop, Go) = Pr(Go, Stop) = 1/2.
c.f. a pair of mixed strategies
Better solution
A traffic signal: Pr((Red, Green)) = Pr((Green, Red)) = 1/2, and
both players agree: Red means Stop, Green means Go.
After they both see the traffic signal, the players have no incentive to
deviate from the agreed actions.
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If we have x ∈ ∆Sm and y ∈ ∆Sn , then choosing the two actions (R, C )
independently gives Pr(R = i, C = j) = xi yj .
In the traffic signal example, we cannot have Pr(Stop, Go) > 0 and
Pr(Go, Stop) > 0 without Pr(Go, Go) > 0.
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Correlated equilibrium
Topics 2
Definition
A correlated strategy pair for a two-player game with payoff matrices A
and B is a correlated equilibrium if
1 for all i, i 0 ∈ S , Pr(R = i) > 0 ⇒ E [a
1
i,C |R = i] ≥ E ai 0 ,C |R = i .
2 for all j, j 0 ∈ S , Pr(C = j) > 0 ⇒ E [b
2
R,j |C = j] ≥ E bR,j 0 |C = j .
c.f. a Nash equilibrium
General-sum games, Nash equilibria.
Two-player: payoff matrices, dominant strategies, safety strategies.
Multiplayer: Utility functions, Nash’s Theorem
Congestion games and potential games
Every potential game has a pure Nash equilibrium
Every congestion game is a potential game
Other equilibrium concepts
Evolutionarily stable strategies
Correlated equilibrium
A strategy profile (x, y) ∈ ∆Sm × ∆Sn is a Nash equilibrium iff
1 for all i, i 0 ∈ S , Pr(R = i) > 0 ⇒ E [a
1
i,C ] ≥ E ai 0 ,C .
2 for all j, j 0 ∈ S , Pr(C = j) > 0 ⇒ E [b
2
R,j ] ≥ E bR,j 0 .
Price of anarchy
When R and C are independent, these expectations and the conditional
expectations are identical.
Thus, a Nash equilibrium is a correlated equilibrium.
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
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Braess’s paradox
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Price of anarchy
Example: before
Definition
For a routing problem, define
price of anarchy =
Example: after
average travel time in worst Nash equilibrium
.
minimal average travel time
The minimum is over all flows.
The flow minimizing average travel time is the socially optimal flow.
The price of anarchy reflects how much average travel time can
decrease in going from a Nash equilibrium flow (where all individuals
choose a path to minimize their travel time) to a prescribed flow.
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(Karlin and Peres, 2016)
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Price of anarchy
Price of anarchy
Example: price of anarchy = 4/3
Example: Nash equals socially optimal
(Karlin and Peres, 2016)
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Price of anarchy
Price of anarchy
Traffic flow
A flow from source s to destination t in a directed graph is a mixture
of paths from s to t, with mixture weight fP for path P.
Theorem
1
For linear latency functions, the price of anarchy is 1.
2
For affine latency functions, the price of anarchy is no more than 4/3.
We write the flow on an edge e as
X
Fe =
fP .
P3e
Latency on an edge e is a non-decreasing function of Fe , written
`e (Fe ).
P
Latency on a path P is the total latency, LP (f ) = e∈P `e (Fe ).
P
P
Average latency is L(f ) = P fP LP (f ) = e Fe `e (Fe ).
A flow f is a Nash equilibrium if, for all P and P 0 , if fP > 0,
LP (f ) ≤ LP 0 (f ).
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The impact of adding edges
Classes of latency functions
Price of anarchy
Suppose we allow latency functions from some class L.
For example, we have considered
Theorem
Consider a network G with a Nash equilibrium flow fG and average latency
LG (fG ), and a network H with additional roads added. Suppose that the
price of anarchy in H is no more than α. Then any Nash equilibrium flow
fH has average latency LH (fH ) ≤ αLG (fG ).
L = {x 7→ ax : a ≥ 0} ,
What about
L = {x 7→ ax + b : a, b ≥ 0} ,
L=
(
x 7→
X
d
)
ad x d : ad ≥ 0 ?
We’ll insist that latency functions are non-negative and
non-decreasing.
It turns out that the price of anarchy in an arbitrary network with
latency functions chosen from L is at most the price of anarchy in a
certain small network with these latency functions: a Pigou network.
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Pigou networks and price of anarchy
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Pigou networks and price of anarchy
A Pigou network
Theorem
Define the Pigou price of anarchy as the price of anarchy for this network
with latency function ` and total flow r :
αr (`) =
r `(r )
.
minx≥0 (x`(x) + (r − x)`(r ))
For any network with latency functions from L and total flow 1, the price
of anarchy is no more than
max max αr (`).
0≤r ≤1 `∈L
(Karlin and Peres, 2016)
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Topics 2
Topics 3
Cooperative games
General-sum games, Nash equilibria.
Transferable versus nontransferable utility
Two-player transferable utility cooperative games
Two-player: payoff matrices, dominant strategies, safety strategies.
Multiplayer: Utility functions, Nash’s Theorem
Cooperative strategy, threat strategies, disagreement point, final payoff
vector
Congestion games and potential games
Two-player nontransferable utility cooperative games
Every potential game has a pure Nash equilibrium
Every congestion game is a potential game
Bargaining problems
Nash’s bargaining axioms, the Nash bargaining solution
Other equilibrium concepts
Multi-player transferable utility cooperative games
Evolutionarily stable strategies
Correlated equilibrium
Characteristic function
Gillies’ core
Shapley’s axioms, Shapley’s Theorem
Price of anarchy
Designing games
Braess’s paradox
The impact of adding edges
Classes of latencies
Pigou networks
Voting systems.
Voting rules, ranking rules
Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem
Properties of voting rules
Instant runoff voting, Borda count, positional voting rules
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Cooperative versus noncooperative games
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Cooperative versus noncooperative games
Cooperative games
Noncooperative games
Players can make binding agreements.
Players play their strategies simultaneously.
e.g.: prisoner’s dilemma.
Both players gain from an enforceable agreement not to confess.
They might communicate (or see a common signal; e.g., a traffic
signal), but there’s no enforced agreement.
Two types:
Transferable utility The players agree what strategies to play and
what additional side payments are to be made.
Nontransferable utility The players choose a joint strategy, but there
are no side payments.
Natural solution concepts:
Nash equilibrium, correlated equilibrium.
No improvement from unilaterally deviating.
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TU cooperative games
TU cooperative games
Feasible payoffs (TU)
Payoff vectors
The set of payoff vectors that the two players can achieve is called the
feasible set.
A feasible payoff vector (v1 , v2 ) is Pareto optimal if the only feasible
payoff vector (v10 , v20 ) with v10 ≥ v1 and v20 ≥ v2 is (v10 , v20 ) = (v1 , v2 ).
(3,6)
Payoffs
With transferable utility, the players can choose any payoff vector in
the convex hull of the set of lines
1
2
(5,5)
1
(2,2)
(4,3)
2
(6,2)
(3,6)
3
(1,2)
(5,5)
{(aij − p, bij + p) : i ∈ {1, . . . , m}, j ∈ {1, . . . , n}, p ∈ R}.
(4,3)
(1,2)
(2,2)
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TU cooperative games
TU cooperative games
Negotiation
Solving two-player TU games
Players negotiate a joint strategy and a side payment.
Since they are rational, they will agree to play a Pareto optimal payoff
vector.
Players might make threats (and counter-threats) to justify their
desired payoff vectors.
1
Find the cooperative strategy (i0 , j0 ) with
ai0 j0 + bi0 j0 = maxi,j (aij + bij ).
2
For the zero-sum game with payoff matrix A − B, find the optimal
strategies x∗ , y∗ (threat strategies) and the value δ = x∗> (A − B)y∗ .
The disagreement point is (d1 , d2 ) = x∗> Ay∗ , x∗> By∗
3
If agreement is not reached, they could carry out their threats.
4
But reaching agreement gives higher utility, so the threats are only
relevant to choosing a reasonable side payment.
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(6,2)
5
The final payoff vector is
(a∗ , b ∗ ) = ((σ − d2 + d1 )/2, (σ − d1 + d2 )/2), with σ = ai0 j0 + bi0 j0 .
The payment from Player I to Player II is ai0 j0 − a∗ .
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TU cooperative games
Topics 3
Cooperative games
Key ideas
Transferable versus nontransferable utility
Two-player transferable utility cooperative games
Cooperative strategy: biggest entry in A + B.
Disagreement point: solution to zero-sum game with payoff matrix
A − B.
Cooperative strategy, threat strategies, disagreement point, final payoff
vector
Two-player nontransferable utility cooperative games
Final payoff vector: midpoint of the Pareto optimal region.
Bargaining problems
Nash’s bargaining axioms, the Nash bargaining solution
Multi-player transferable utility cooperative games
Characteristic function
Gillies’ core
Shapley’s axioms, Shapley’s Theorem
(3,6)
(5,5)
Designing games
Voting systems.
(4,3)
(1,2)
(2,2)
Voting rules, ranking rules
Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem
Properties of voting rules
Instant runoff voting, Borda count, positional voting rules
(6,2)
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Nontransferable utility cooperative games
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Nash Bargaining Model for NTU games
Ingredients of a bargaining problem
Feasible payoffs (NTU)
1
2
(3,6)
Payoffs
1
2
2
(6,2)
(3,6)
3
(1,2)
(5,5)
(4,3)
(1,2)
(2,2)
A disagreement point d = (d1 , d2 ) ∈ R2 .
Think of the disagreement point as the utility that the players get from
walking away and not playing the game.
(And we’ll assume every x ∈ S has x1 ≥ d1 , x2 ≥ d2 , with strict
inequalities for some x ∈ S.)
(5,5)
1
(2,2)
(4,3)
A compact, convex feasible set S ⊂ R2 .
(6,2)
Definition
A solution to a bargaining problem is a function F that takes
a feasible set S and a disagreement point d and returns an agreement
point a = (a1 , a2 ) ∈ S.
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Nash Bargaining Model for NTU games
Nash Bargaining Model for NTU games
Nash’s bargaining axioms
Nash’s bargaining axioms
1
2
3
4
Pareto optimality: the only feasible payoff vector (v1 , v2 ) with v1 ≥ a1
and v2 ≥ a2 is (v1 , v2 ) = (a1 , a2 ).
Symmetry: If both (x, y ) ∈ S implies (y , x) ∈ S and d1 = d2 then
a1 = a2 .
Affine covariance: For any affine transformation
Ψ(x1 , x2 ) = (α1 x1 + β1 , α2 x2 + β2 ) with α1 , α2 > 0 and any S and d,
F (Ψ(S), Ψ(d)) = Ψ(F (S, d)).
Independence of irrelevant attributes: For two bargaining problems
(R, d) and (S, d), if R ⊂ S and F (S, d) ∈ R, then
F (R, d) = F (S, d).
Pareto optimality: The agreement point shouldn’t be dominated by
another point for both players. (Criticism: why should one player care
if the agreement point is only dominated for the other player?)
Symmetry: This is about fairness: if nothing distinguishes the players,
the solution should be similarly symmetric.
Affine covariance: Changing the units (or a constant offset) of the
utilities should not affect the outcome of bargaining.
Independence of irrelevant attributes: This assumes that all of the
threats the players might make have been accounted for in the
disagreement point.
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Nash Bargaining Model for NTU games
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Topics 3
Cooperative games
Transferable versus nontransferable utility
Two-player transferable utility cooperative games
Theorem
There is a unique function F satisfying Nash’s bargaining axioms.
It is the function that takes S and d and returns the unique solution to the
optimization problem
max
(x1 ,x2 )
subject to
(x1 − d1 )(x2 − d2 )
Cooperative strategy, threat strategies, disagreement point, final payoff
vector
Two-player nontransferable utility cooperative games
Bargaining problems
Nash’s bargaining axioms, the Nash bargaining solution
Multi-player transferable utility cooperative games
Characteristic function
Gillies’ core
Shapley’s axioms, Shapley’s Theorem
x1 ≥ d1
x2 ≥ d2
Designing games
(x1 , x2 ) ∈ S.
Voting systems.
Voting rules, ranking rules
Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem
Properties of voting rules
Instant runoff voting, Borda count, positional voting rules
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Multiplayer TU cooperative games
The core
Characteristic function
Gillies’ core
We focus on the formation of coalitions in multiplayer TU games.
Efficiency
Pn
= v ({1, . . . , n}).
P
Stability For all S ⊆ {1, . . . , n}, i∈S ψi (v ) ≥ v (S).
Define a characteristic function: for each subset S of players, v (S) is
the total value that would be available to be split by that subset of
players, no matter what the other players do.
i=1 ψi (v )
These seem reasonable properties:
(Called a coalitional form game, versus a strategic form game.)
Allocations
Define an allocation function ψ as a map from a characteristic function v
for n players to a vector ψ(v ) ∈ Rn .
This is the payoff that is allocated to the n players.
1
The total payoff gets allocated. Anything less is irrational.
2
A coalition gets allocated at least the payoff it can obtain on its own.
But there might be no allocation, one allocation, or many allocations in
the core!
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Shapley value
Shapley value
Shapley’s Theorem
Shapley axioms
Efficiency
The following allocation uniquely satisfies Shapley’s axioms:
ψi (v ) = Eπ φi (v , π),
Pn
i=1 ψi (v ) = v ({1, . . . , n}).
Symmetry If, for all S ⊂ {1, . . . , n} not containing i, j,
v (S ∪ {i}) = v (S ∪ {j}), then ψi (v ) = ψj (v ).
Dummy If for all S ⊂ {1, . . . , n}, v (S ∪ {i}) = v (S), then ψi (v ) = 0.
Additivity ψi (v + u) = ψi (v ) + ψi (u).
Shapley’s Theorem
Shapley’s axioms uniquely determine the allocation ψ.
We call the unique allocation ψ(v ) the Shapley value of the players in the
game defined by the characteristic function v .
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where the expectation is over uniformly chosen permutations π on
{1, . . . , n}, and
φi (v , π) = v π {1, . . . , π −1 (i)} − v π {1, . . . , π −1 (i) − 1} .
Example: For the identity permutation, π(i) = i,
φi (v , π) = v ({1, . . . , i}) − v ({1, . . . , i − 1}) ,
which is how much value i adds to {1, . . . , i − 1}.
And for a random π, φi (v , π) is how much value i adds to the random set
π {1, . . . , π −1 (i) − 1} .
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Shapley value
Topics 3
Example: Junta game (J-veto game)
(
1 if J ⊆ S,
wJ (S) = 1[J ⊆ S] =
0 otherwise.
Cooperative games
ψi (wJ ) = 1 [i ∈ J]
Transferable versus nontransferable utility
Two-player transferable utility cooperative games
1
.
|J|
Cooperative strategy, threat strategies, disagreement point, final payoff
vector
Two-player nontransferable utility cooperative games
Bargaining problems
Nash’s bargaining axioms, the Nash bargaining solution
Lemma [Characteristic functions as junta games]
Multi-player transferable utility cooperative games
We can write any v as a unique linear combination of wJ ’s.
Characteristic function
Gillies’ core
Shapley’s axioms, Shapley’s Theorem
Computing Shapley value
Designing games
We can P
always write the characteristic function as
v (S) = J cJ wJ (S), where the sum is over nonempty subsets
J ⊆ {1, . . . , n}.
Voting systems.
Voting rules, ranking rules
Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem
Properties of voting rules
Instant runoff voting, Borda count, positional voting rules
We know that ψi (wJ ) = 1[i ∈ J]/|J|.
This is often an easy approach to computing the Shapley value.
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Designing games and mechanisms
Social choice and Voting
We aim to design the rules of a game so that the outcomes have certain
desired properties.
Electing a president
Elections
Consistent with voters’ rankings
Fair (symmetric)
Auctions
Maximize revenue for the seller
Pareto efficiency
Calibrated (revealing bidders’ values)
Tournaments
Suppose there are two candidates for president, and all voters have a
preference.
How do we design an election to decide between the two candidates?
Voters vote; the candidate with the most votes wins.
The candidate that wins is the choice of at least half of the voters.
Voters never have an incentive to vote against their preferences.
The best team is most likely to win
Players have an incentive to compete
Things aren’t as simple with three candidates.
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Topics 3
How do we formulate a voting mechanism?
Cooperative games
Transferable versus nontransferable utility
Two-player transferable utility cooperative games
Cooperative strategy, threat strategies, disagreement point, final payoff
vector
Questions
1
How do we model voters’ preferences?
2
How do voters’ express their preferences?
3
How do we combine that information?
Two-player nontransferable utility cooperative games
Bargaining problems
Nash’s bargaining axioms, the Nash bargaining solution
Multi-player transferable utility cooperative games
Characteristic function
Gillies’ core
Shapley’s axioms, Shapley’s Theorem
We will distinguish two outcomes:
A single winner (“voting rule”)
Designing games
A ranking of all candidates (“ranking rule”)
Voting systems.
Voting rules, ranking rules
Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem
Properties of voting rules
Instant runoff voting, Borda count, positional voting rules
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Topics 3
Voting and Ranking
Cooperative games
Transferable versus nontransferable utility
Two-player transferable utility cooperative games
Assumptions
Cooperative strategy, threat strategies, disagreement point, final payoff
vector
Two-player nontransferable utility cooperative games
There is a set Γ of candidates.
Voter i has a preference relation “i ” defined on candidates that is:
1
Bargaining problems
Nash’s bargaining axioms, the Nash bargaining solution
2
Multi-player transferable utility cooperative games
Complete: ∀A 6= B ∈ Γ, A i B or B i A.
Transitive: ∀A, B, C ∈ Γ, A i B and B i C implies A i C .
Definitions
Characteristic function
Gillies’ core
Shapley’s axioms, Shapley’s Theorem
A voting rule f maps a preference profile π = (1 , . . . , n ) to a
winner from Γ.
Designing games
A ranking rule R maps a preference profile π = (1 , . . . , n ) to a
social ranking “” on Γ, which is another complete, transitive
preference relation.
Voting systems.
Voting rules, ranking rules
Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem
Properties of voting rules
Instant runoff voting, Borda count, positional voting rules
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Properties of ranking rules
Properties of ranking rules
Strategically vulnerable
Unanimity
A ranking rule R has the
unanimity property if, for all i,
A i B, then = R(1 , . . . , n )
satisfies A B.
If all voters prefer candidate
A over B, candidate A
should be ranked above B.
A ranking rule R is strategically
vulnerable if, for some preference
profile (1 , . . . , n ), some voter
i and some candidates A, B ∈ Γ,
:= R(1 , . . . , i , . . . , n ),
0 := R(1 , . . . , 0i , . . . , n ),
A i B, B A, but A 0 B.
This means that Voter i has
a preference relation i , but
by stating an alternative
preference relation 0i , it can
swap the ranking rule’s
preference between A and B
to make it consistent with
i .
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Properties of ranking rules
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Properties of ranking rules
Theorem
Any ranking rule R that violates IIA is strategically vulnerable.
Independence of irrelevant
alternatives (IIA)
Consider two different voter
preference profiles (1 , . . . , n )
and (01 , . . . , 0n ), and define
= R(1 , . . . , n ) and
0 = R(01 , . . . , 0n ). For
A, B ∈ Γ, if, for all i, A i B iff
A 0i B, then A B iff A 0 B.
Definition
The ranking rule’s relative
rankings of candidates A and
B should depend only on the
voters’ relative rankings of
these two candidates.
A ranking rule R is a dictatorship if there is a voter i ∗ such that, for any
preference profile (1 , . . . , n ), = R(1 , . . . , n ) has A B iff A i ∗ B.
Arrow’s Impossibility Theorem
For |Γ| ≥ 3, any ranking rule R that satisfies both IIA and unanimity is a
dictatorship.
Thus, any ranking rule R that satisfies unanimity and is not strategically
vulnerable is a dictatorship.
Hence, strategic vulnerability is inevitable.
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Topics 3
Strategic vulnerability
Cooperative games
Transferable versus nontransferable utility
Two-player transferable utility cooperative games
Cooperative strategy, threat strategies, disagreement point, final payoff
vector
Two-player nontransferable utility cooperative games
Bargaining problems
Nash’s bargaining axioms, the Nash bargaining solution
Multi-player transferable utility cooperative games
Recall: Ranking
Voting
A ranking rule R is strategically
vulnerable if, for some preference
profile (1 , . . . , n ), some voter
i and some candidates A, B ∈ Γ,
A voting rule f is strategically
vulnerable if, for some preference
profile (1 , . . . , n ), some voter i
and some candidates A, B ∈ Γ,
:= R(1 , . . . , i , . . . , n ),
Characteristic function
Gillies’ core
Shapley’s axioms, Shapley’s Theorem
0 := R(1 , . . . , 0i , . . . , n ),
A i B, B A, but A 0 B.
Designing games
Voting systems.
π := (1 , . . . , i , . . . , n ),
π 0 := (1 , . . . , 0i , . . . , n ),
A i B, B = f (π), but A = f (π 0 ).
Voter i, by incorrectly reporting preferences, can change the outcome
to match his true preferences.
Voting rules, ranking rules
Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem
Properties of voting rules
Instant runoff voting, Borda count, positional voting rules
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Dictatorship
Another impossibility theorem
Recall: Arrow’s Impossibility Theorem
For |Γ| ≥ 3, any ranking rule R that satisfies unanimity and is not
strategically vulnerable is a dictatorship.
Ranking
Voting
A ranking rule R is a dictatorship
if there is a voter i ∗ such that,
for any preference profile
(1 , . . . , n ), = R(1 , . . . , n )
has A B iff A i ∗ B.
A voting rule f is a dictatorship if
there is a voter i ∗ such that, for
any preference profile
(1 , . . . , n ), A = f (1 , . . . , n )
iff for all B 6= A, A i ∗ B.
Voter i ∗ determines the outcome.
A voting rule f maps from the voters’ preference profile π to a winner
in Γ.
We say f is onto Γ if, for all candidates A ∈ Γ, there is a π satisfying
f (π) = A.
If f is not onto Γ, some candidate is excluded from winning.
Gibbard-Satterthwaite Theorem
For |Γ| ≥ 3, any voting rule f that is onto Γ and is not strategically
vulnerable is a dictatorship.
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Topics 3
Properties of voting systems
Cooperative games
Symmetry: Permuting voters does not affect the outcome.
Transferable versus nontransferable utility
Two-player transferable utility cooperative games
Cooperative strategy, threat strategies, disagreement point, final payoff
vector
Two-player nontransferable utility cooperative games
Monotonicity: Changing one voter’s preferences by promoting
candidate A without changing any other preferences should not
change the outcome from A winning to A not winning.
Condorcet winner criterion: If a candidate is majority-preferred in
pairwise comparisons with all other candidates, then that candidate
wins.
Bargaining problems
Nash’s bargaining axioms, the Nash bargaining solution
Multi-player transferable utility cooperative games
Characteristic function
Gillies’ core
Shapley’s axioms, Shapley’s Theorem
Condorcet loser criterion: If a candidate is preferred by a minority
of voters in pairwise comparisons with all other candidates, then that
candidate should not win.
Designing games
Voting systems.
Smith criterion The winner always comes from the Smith set (the
smallest nonempty set of candidates that are majority-preferred in
pairwise comparisons with any candidate outside the set).
Voting rules, ranking rules
Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem
Properties of voting rules
Instant runoff voting, Borda count, positional voting rules
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Properties of voting systems
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Topics 3
Cooperative games
Reversal symmetry: If A wins for some voter preference profile, A
does not win when the preferences of all voters are reversed.
Cancellation of ranking cycles: If a set of |Γ| voters have
preferences that are cyclic shifts of each other (e.g., A 1 B 1 C ,
B 2 C 2 A, C 3 A 3 B), then removing these voters does not
affect the outcome.
Cancellation of opposing rankings: If two voters have reversed
preferences, then removing these voters does not affect the outcome.
Consistency: If A wins for voter preference profiles π and π 0 , A also
wins when these voter preference profiles are combined.
Participation: If A wins for some voter preference profile, then
adding a voter with A B does not change the winner from A to B.
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Transferable versus nontransferable utility
Two-player transferable utility cooperative games
Cooperative strategy, threat strategies, disagreement point, final payoff
vector
Two-player nontransferable utility cooperative games
Bargaining problems
Nash’s bargaining axioms, the Nash bargaining solution
Multi-player transferable utility cooperative games
Characteristic function
Gillies’ core
Shapley’s axioms, Shapley’s Theorem
Designing games
Voting systems.
Voting rules, ranking rules
Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem
Properties of voting rules
Instant runoff voting, Borda count, positional voting rules
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Properties of voting systems
Properties of voting systems
Instant runoff voting
Voters provide a ranking of the candidates.
If only one candidate remains, return that candidate.
Otherwise:
1
2
3
Properties of instant runoff voting
Monotonicity? (Changing one voter’s preferences by promoting
candidate A without changing any other preferences does not change
the outcome from A winning to A not winning.) No.
Eliminate the candidate that is top-ranked by the fewest voters.
Drop that candidate’s preferences from voters’ rankings.
Use instant runoff voting on the remaining candidates with the
reassigned preferences.
Properties of instant runoff voting
Symmetry? (Permuting voters does not affect the outcome.) Yes.
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Properties of voting systems
Properties of voting systems
Properties of instant runoff voting
Properties of instant runoff voting
Condorcet winner criterion? (If a candidate is majority-preferred in
pairwise comparisons with all other candidates, then that candidate
wins.) No.
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Condorcet loser criterion? (If a candidate is preferred by a minority
of voters in pairwise comparisons with all other candidates, then that
candidate should not win.) Yes.
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Properties of voting systems
Properties of voting systems
Properties of instant runoff voting
Properties of instant runoff voting
Smith criterion? (The winner always comes from the Smith set—the
smallest nonempty set of candidates that are majority-preferred in
pairwise comparisons with any candidate outside the set.) No.
(Notice that a rule that violates the Condorcet winner criterion violates
the Smith criterion for some preference profile with a singleton Smith set.)
Reversal symmetry? (If A wins for some voter preference profile, A
does not win when the preferences of all voters are reversed.) No.
Cancellation of ranking cycles? No.
Cancellation of opposing rankings? No.
Consistency? No.
Participation? No.
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Topics 3
Properties of voting systems
Cooperative games
Transferable versus nontransferable utility
Two-player transferable utility cooperative games
Cooperative strategy, threat strategies, disagreement point, final payoff
vector
Borda count
Two-player nontransferable utility cooperative games
Bargaining problems
Nash’s bargaining axioms, the Nash bargaining solution
Voters rank candidates from 1 to N (where N = |Γ|).
A candidate that is ranked in ith position is assigned N − i + 1 points.
Multi-player transferable utility cooperative games
Characteristic function
Gillies’ core
Shapley’s axioms, Shapley’s Theorem
The candidate with the largest total wins.
Designing games
Voting systems.
Voting rules, ranking rules
Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem
Properties of voting rules
Instant runoff voting, Borda count, positional voting rules
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Properties of voting systems
Properties of voting systems
Positional voting rules
Properties of positional voting rules
Define a1 ≥ a2 ≥ · · · ≥ aN .
Symmetry? Yes.
For each candidate, assign ai points for each voter that assigns that
candidate rank i.
Monotonicity? Yes.
The candidate with the largest total wins.
Cancellation of ranking cycles? Yes.
Condorcet winner criterion? No.
Consistency? Yes.
e.g., Borda count: N, N − 1, . . . , 1.
e.g., Plurality: 1, 0, . . . , 0.
e.g., Approval voting: 1, 1, . . . , 1, 0, . . . , 0.
Participation? Yes.
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Topics 3
Cooperative games
Transferable versus nontransferable utility
Two-player transferable utility cooperative games
Cooperative strategy, threat strategies, disagreement point, final payoff
vector
Two-player nontransferable utility cooperative games
Bargaining problems
Nash’s bargaining axioms, the Nash bargaining solution
Multi-player transferable utility cooperative games
Characteristic function
Gillies’ core
Shapley’s axioms, Shapley’s Theorem
Designing games
Voting systems.
Voting rules, ranking rules
Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem
Properties of voting rules
Instant runoff voting, Borda count, positional voting rules
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