Stat155 Game Theory Lecture 27: Final Review Final Exam Topics 1

Final Exam

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 Peter Bartlett communicate). Answer each question in the space provided.”

December 6, 2016

1 / 127 2 / 127 Topics 1 Topics 2

Combinatorial games Positions, moves, terminal positions, impartial/partisan, progressively General-sum games, Nash equilibria. bounded Two-player: payoff matrices, dominant strategies, safety strategies. Progressively bounded impartial and partisan games Multiplayer: Utility functions, Nash’s Theorem The sets N and P Congestion games and potential games Theorem: Someone can win Every potential game has a pure Nash equilibrium Examples: Subtraction, Chomp, Nim, Rims, Staircase Nim, Hex Every congestion game is a potential game Zero sum games Other equilibrium concepts Payoff matrices, pure and mixed strategies, safety strategies Evolutionarily stable strategies Von Neumann’s minimax theorem Solving two player zero-sum games Correlated equilibrium Saddle points Price of anarchy Equalizing strategies Braess’s paradox Solving2 2 games × The impact of adding edges Dominated strategies Classes of latencies 2 n and m 2 games × × Pigou networks Principle of indifference Symmetry: Invariance under permutations

3 / 127 4 / 127 Topics 3 Deﬁnitions: Combinatorial games

Cooperative games Transferable versus nontransferable utility A combinatorial game has: Two-player transferable utility cooperative games Two players, Player I and Player II. Cooperative strategy, threat strategies, disagreement point, final payoff vector A set of positions X . Two-player nontransferable utility cooperative games For each player, a set of legal moves between positions, that is, a set Bargaining problems of ordered pairs, (current position, next position): 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 Designing games Players alternately choose moves. Voting systems. Play continues until some player cannot move. Voting rules, ranking rules Normal play: the player that cannot move loses the game. Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem Properties of voting rules Instant runoff voting, Borda count, positional voting rules

5 / 127 6 / 127 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 . Theorem A partisan game has diﬀerent sets of legal moves for the players. In a progressively bounded impartial A terminal position for a player has no legal move to another position. combinatorial game under normal x is terminal for player I if there is no y X with( x, y) MI . play, X = N P. ∈ ∈ ∪ A combinatorial game is progressively bounded if, for every starting That is, from any initial position, one position x X , there is ﬁnite bound on the number of moves before of the players has a winning strategy. 0 ∈ the game ends. (That is, if B(x) denotes the maximum number of Proof: induction on number of moves before the game ends, then B(x) < .) ∞ moves until the end. 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.

7 / 127 8 / 127 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( x ,..., x ) is in P iﬀ N: Some move leads to P (hence cannot contain terminal positions). 1 k the Nim-sum of its components is0. The Nim-sum of x = (x ,..., x ) is written x x x . 1 k 1 ⊕ 2 ⊕ · · · ⊕ k 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 := (x ,..., x ): x x = 0 is P: { 1 k 1 ⊕ · · · ⊕ k } Z must lead to Z c ; from Z c , there is a move into Z.

9 / 127 10 / 127 Partisan Games Topics 1

Combinatorial games Positions, moves, terminal positions, impartial/partisan, progressively Recall: bounded Progressively bounded impartial and partisan games An impartial game has the same set of legal moves for both players: The sets N and P MI = MII . Theorem: Someone can win A partisan game has different sets of legal moves for the players. Examples: Subtraction, Chomp, Nim, Rims, Staircase Nim, Hex Zero sum games Theorem Payoff matrices, pure and mixed strategies, safety strategies Consider a progressively bounded partisan Von Neumann’s minimax theorem combinatorial game under normal play, with no ties Solving two player zero-sum games allowed. From any initial position, one of the players Saddle points Equalizing strategies has a winning strategy. Solving2 2 games × Dominated strategies 2 n and m 2 games × × Principle of indifference Symmetry: Invariance under permutations

11 / 127 12 / 127 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. m n The payoff matrix A R × represents the payoff to Player I: ∈ A mixed strategy for Player I is a vector

a11 a12 a1n ··· x1 a a a 21 22 2n x m A =  . ···. .  2 m . . . x =  .  ∆m := x R : xi 0, xi = 1 . . . . . ∈ ∈ ≥   . ( i=1 ) am1 am2 amn   X  ···  xm     If Player I chooses i and Player II chooses j, the payoff to Player I is   a and the payoff to Player II is a . ij − ij The sum of the payoff to Player I and the payoff to Player II is0.

13 / 127 14 / 127 Two-player zero-sum games Two-player zero-sum games

The expected payoﬀ to Player I when Player I plays mixed strategy A mixed strategy for Player II is a vector x ∆m and Player II plays mixed strategy y ∆n is ∈ ∈ m n y1 n EI x EJ y aIJ = xi aij yj y2 ∼ ∼   n i=1 j=1 y = . ∆n := y R : yi 0, yi = 1 . X X . ∈ ( ∈ ≥ )   Xi=1 = x>Ay yn   a a a y   11 12 ··· 1n 1 a21 a22 a2n y2 A pure strategy is a mixed strategy where one entry is1 and the = x1 x2 xm  . ···. .   .  ··· . .. . . others0. (This is a canonical basis vector ei .)     a a a  y   m1 m2 ··· mn  n    

15 / 127 16 / 127 Two-player zero-sum games Two-player zero-sum games

Von Neumann’s Minimax Theorem A safety strategy for Player I is an x ∆ that satisﬁes m n ∗ m For any two-person zero-sum game with payoﬀ matrix A R × , ∈ ∈

min x∗>Ay = max min x>Ay. max min x>Ay = min max x>Ay. y ∆n x ∆m y ∆n x ∆m y ∆n y ∆n x ∆m ∈ ∈ ∈ ∈ ∈ ∈ ∈

This mixed strategy maximizes the worst case expected gain for Player I. We call the optimal expected payoﬀ the value of the game,

Similarly, a safety strategy for Player II is a y ∗ ∆n that satisfies V := max min x>Ay = min max x>Ay. x ∆m y ∆n y ∆n x ∆m ∈ ∈ ∈ ∈ ∈ max x>Ay ∗ = min max x>Ay. x ∆m y ∆n x ∆m LHS: Player I plays x ∆m first, then Player II responds with y ∆n. ∈ ∈ ∈ ∈ ∈ RHS: Player II plays y ∆ first, then Player I responds with x ∆ . ∈ n ∈ m Notice that we should always prefer to play last: 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. 17 / 127 18 / 127 Two-player zero-sum games Topics 1

Combinatorial games Von Neumann’s Minimax Theorem Positions, moves, terminal positions, impartial/partisan, progressively m n For any two-person zero-sum game with payoff matrix A R × , ∈ bounded Progressively bounded impartial and partisan games max min x>Ay = min max x>Ay. x ∆m y ∆n y ∆n x ∆m Zero sum games ∈ ∈ ∈ ∈ Payoff matrices, pure and mixed strategies, safety strategies Von Neumann’s minimax theorem Solving two player zero-sum games Safety strategies are optimal strategies: Saddle points For safety strategies x∗ for Player I and y ∗ for Player II, Equalizing strategies Solving2 2 games × min x∗>Ay = max x>Ay ∗ = x∗>Ay ∗ = V . Dominated strategies y ∆n x ∆m 2 n and m 2 games ∈ ∈ × × Principle of indifference Symmetry: Invariance under permutations

19 / 127 20 / 127 Saddle points Saddle points

Deﬁnition

A pair( i ∗, j∗) 1,..., m 1,..., n is a saddle point for a payoﬀ m∈n { } × { } matrix A R × if Theorem ∈ m n If( i ∗, j∗) is a saddle point for a payoﬀ matrix A R × , then max aij = ai j = min ai j . ∈ i ∗ ∗ ∗ j ∗ ei ∗ is an optimal strategy for Player I,

ej∗ is an optimal strategy for Player II, and If Player I plays i and Player II plays j , neither player has an ∗ ∗ the value of the game is ai ∗j∗ . incentive to change. Think of these as locally optimal strategies for the players. They are also globally optimal strategies.

21 / 127 22 / 127 2 2 games Dominated pure strategies × How to solve a2 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

Equalizing strategies satisfy: A pure strategy ei for Player I is dominated by ei 0 in payoﬀ matrix A if, for all j 1,..., n , ∈ { } x1a11 + (1 x1)a21 = x1a12 + (1 x1)a22, a a . − − ij ≤ i 0j y a + (1 y )a = y a + (1 y )a . 1 11 − 1 12 1 21 − 1 22 Solving gives

a21 a22 x1 = − , a21 a22 + a12 a11 − a a − y = 12 − 22 . 1 a a + a a 12 − 22 21 − 11 23 / 127 24 / 127 Solving2 n games Principle of indifference × Payoff matrix Theorem m n Suppose a game with payoff matrix A R × has value V . If x ∆m 2315 ∈ ∈ and y ∆ are optimal strategies for Players I and II, then 4160 n ∈ m n for all j, x a V , for all i, a y V , l lj ≥ il l ≤ The maximum occurs at the Xl=1 Xl=1 intersection of the lines m n corresponding to columns 2 and 3. if yj > 0, xl alj = V , if xi > 0, ail yl = V . l=1 l=1 The optimal strategy for Player I is X X x = (5/7, 2/7). Then Player II is indifferent between This means that if one player is playing optimally, any action that has columns 2 and 3. positive weight in the other player’s optimal mixed strategy is a The optimal strategy for Player II suitable response.

(Ferguson, 2014) involves only Columns 2 and 3. We It implies that any mixture of these “active actions” is also a suitable can ﬁnd it by solving a2 2 game. response. × 25 / 127 26 / 127 Using the principle of indiﬀerence Example

Diagonal payoﬀ matrix

Solving linear systems a11 00 A = 0 a 0 Suppose that we have a payoff matrix A and we suspect that an  22  optimal strategy for Player I has certain components positive, say 00 a33 x1 > 0, x3 > 0.   Then we can solve the corresponding “indifference equalities” to find The aii are all positive, so we suspect that all xi , yi > 0 for the y, say optimal strategies. Solve n n a1l yl = V , a3l yl = V . V V V > x>A = VVV : x = y = a a a , l=1 l=1 11 22 33 X X 1 V = . 1/a11 + 1/a22 + 1/a33

27 / 127 28 / 127 Topics 1 Symmetry

Combinatorial games Positions, moves, terminal positions, impartial/partisan, progressively Definition bounded m n A game with payoff matrix A R × is invariant under a permutation πx Progressively bounded impartial and partisan games ∈ on 1,..., m if there is a permutation πy on 1,..., n such that, for all Zero sum games { } { } i, j, aij = aπx (i),πy (j). Payoff matrices, pure and mixed strategies, safety strategies Von Neumann’s minimax theorem Solving two player zero-sum games If A is invariant under permutations π and π on 1,..., m , then it 1 2 { } Saddle points is invariant under π1 π2. Equalizing strategies ◦ Solving2 2 games If A is invariant under some set S of permutations, then it is invariant × Dominated strategies under the group G of permutations generated by S (that is, 2 n and m 2 games × × compositions and inverses). Principle of indifference Symmetry: Invariance under permutations

29 / 127 30 / 127 Symmetry Topics 1

Combinatorial games Definition Positions, moves, terminal positions, impartial/partisan, progressively A mixed strategy x ∆ is invariant under a permutation π on ∈ m x bounded 1,..., m if for all i, xi = x . Progressively bounded impartial and partisan games { } πx (i) Zero sum games Payoff matrices, pure and mixed strategies, safety strategies An orbit of a group G of permutations is a set Oi = π(i): π G . { ∈ } Von Neumann’s minimax theorem If a mixed strategy x is invariant under a group G of permutations, Solving two player zero-sum games then for every orbit, x is constant on the orbit. Saddle points Equalizing strategies Solving2 2 games Theorem × Dominated strategies If A is invariant under a group G of permutations, then there are optimal 2 n and m 2 games × × strategies that are invariant under G. Principle of indifference Symmetry: Invariance under permutations

31 / 127 32 / 127 Topics 2 Noncooperative games

General-sum games, Nash equilibria. Two-player: payoﬀ matrices, dominant strategies, safety strategies. Multiplayer: Utility functions, Nash’s Theorem Congestion games and potential games Players play their strategies simultaneously. Every potential game has a pure Nash equilibrium They might communicate (or see a common signal; e.g., a traﬃc Every congestion game is a potential game signal), but there’s no enforced agreement. Other equilibrium concepts Natural solution concepts: Evolutionarily stable strategies Nash equilibrium, correlated equilibrium. Correlated equilibrium No improvement from unilaterally deviating. Price of anarchy Braess’s paradox The impact of adding edges Classes of latencies Pigou networks

33 / 127 34 / 127 General-sum games General-sum games

Dominated pure strategies Notation A pure strategy e for Player I is dominated by e in payoff matrix A if, for A two-person general-sum game is specified by two payoff matrices, i i 0 m n all j 1,..., n , A, B R × . ∈ { } ∈ aij ai j . Simultaneously, Player I chooses i 1,..., m and the Player II ≤ 0 ∈ { } chooses j 1,..., n . Similarly, a pure strategy ej for Player II is dominated by ej0 in payoff ∈ { } matrix B if, for all i 1,..., m , Player I receives payoff aij . ∈ { } Player II receives payoff bij . b b . ij ≤ ij0

35 / 127 36 / 127 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 ) ∆ ∆ is a Nash equilibrium for payoff matrices ∗ m n ∈ ∗m n∗ ∈ × A, B R × 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 ∗ y ∆n x ∆m If Player I plays x and Player II plays y , neither player has an ∈ ∈ ∈ ∗ ∗ incentive to unilaterally deviate. 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 In general-sum games, there might be many Nash equilibria, with conservative. diﬀerent payoﬀ vectors.

37 / 127 38 / 127 Comparing two-player general-sum and zero-sum games Comparing two-player general-sum and zero-sum games

Zero-sum games Zero-sum games 1 A pair of safety strategies is a Nash equilibrium (minimax theorem) 4 If each player has an equalizing mixed strategy 2 Hence, there is always a Nash equilibrium. (that is, x>A = v1> and Ay = v1), 3 If there are multiple Nash equilibria, they form a convex set, and the then this pair of strategies is a Nash equilibrium. expected payoff is identical within that set. (from the principle of indifference) Thus, any two Nash equilibria give the same payoff. General-sum games General-sum games 4 If each player has an equalizing mixed strategy 1 A pair of safety strategies might be unstable. for their opponent’s payoff matrix (Opponent aims to maximize their payoff, not minimize mine.) (that is, x>B = v21> and Ay = v11), 2 There is always a Nash equilibrium (Nash’s Theorem). then this pair of strategies is a Nash equilibrium. 3 There can be multiple Nash equilibria, with different payoff vectors.

39 / 127 40 / 127 Topics 2 Multiplayer general-sum games

General-sum games, Nash equilibria. Notation Two-player: payoff matrices, dominant strategies, safety strategies. A k-person general-sum game is specified by k utility functions, Multiplayer: Utility functions, Nash’s Theorem uj : S1 S2 Sk R. × × · · · × → Congestion games and potential games Player j can choose strategies s S . j ∈ j Every potential game has a pure Nash equilibrium Simultaneously, each player chooses a strategy. Every congestion game is a potential game 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 .

Price of anarchy For s = (s1,..., sk ), let s i denote the strategies without the ith one: − Braess’s paradox The impact of adding edges s i = (s1,..., si 1, si+1,..., sk ). − − Classes of latencies Pigou networks And write( si , s i ) as the full vector. −

41 / 127 42 / 127 Multiplayer general-sum games Multiplayer general-sum games

Definition A sequence( x ,..., x ) ∆ ∆ (called a strategy profile) is a 1∗ k∗ ∈ S1 × · · · × Sk Definition Nash equilibrium for utility functions u1,..., uk if, for each player j 1,..., k , A vector( s ,..., s ) S S is a pure Nash equilibrium for utility ∈ { } 1∗ k∗ ∈ 1 × · · · × k functions u1,..., uk if, for each player j 1,..., k , ∈ { } max uj (xj , x∗ j ) = uj (xj∗, x∗ j ). xj ∆S − − ∈ j max uj (sj , s∗ j ) = uj (sj∗, s∗ j ). sj Sj − − ∈ Here, we define

uj (x) = Es1 x1,...,sk xk uj (s1,..., sk ) 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’ = x (s ) x (s )u (s ,..., s ). 1 1 ··· k k j 1 k strategies. s S ,...,s S 1∈ 1X k ∈ k

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.

43 / 127 44 / 127 Multiplayer general-sum games Multiplayer general-sum games

Nash’s Theorem (1951) Every finite general-sum game has a Nash equilibrium. Theorem Consider a strategy profile x ∆ ∆ . Let Proof Idea (two players) ∈ S1 × · · · × Sk T = s S : x (s) > 0 . The following statements are equivalent. We find an “improvement” map M(x, y) = (ˆx, yˆ), so that i { ∈ i i } 1 x is a Nash equilibrium. 1 xˆ>Ay > x>Ay (orˆx>Ay = x>Ay, if x was a best response to y),

2 For each i, there is a ci such that 2 x>Byˆ > x>By (or x>Byˆ = x>By, if y was a best response to x),

1 For all si Si , ui (si , x i ) ci . No better response outside Ti 3 ∈ − ≤ ←− M is continuous. 2 For si Ti , ui (si , x i ) = ci . Indifferent within Ti ∈ − ←− It’s easy to find a map like this. For instance, take a step in the direction of increasing payoff: Compare with the principle of indifference in the zero-sum case. xˆ = (x + ηAy),ˆ y = (y + ηB x). P∆m P∆n > A Nash equilibrium is a fixed point of M. The existence of a Nash equilibrium follows from Brouwer’s fixed-point theorem.

45 / 127 46 / 127 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 Brouwer’s Fixed-Point Theorem Every potential game has a pure Nash equilibrium Every congestion game is a potential game A continuous map f : K K from a convex, closed, bounded K Rd → ⊆ Other equilibrium concepts has a ﬁxed point, that is, an x K satisfying f (x) = x. ∈ Evolutionarily stable strategies Correlated equilibrium Price of anarchy Braess’s paradox The impact of adding edges Classes of latencies Pigou networks

47 / 127 48 / 127 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 { } A For player i, there is a set Si of strategies that are sets of facilities, s 1,..., m (e.g., paths) ⊆ { } k (1,2,4) (2,3,5) For facility j, there is a cost vector cj R , where cj (n) is the cost of ∈ facility j when it is used by n players.

I, II, III S (1,2,6) T For a sequence s = (s1,..., sn), the utilities of the players are deﬁned by

cost (s) = u (s) = c (n (s)), (2,3,5) (2,4,8) i − i j j j s X∈ i B where n (s) = i : j s is the number of players using facility j. j |{ ∈ i }| Egalitarian: The utilities depend on how many players use each facility, and not on which players use it.

49 / 127 50 / 127 Congestion games Congestion games

Proof We deﬁne a potential function Φ: S1 Sk R as × · · · × →

m nj (s) Φ(s) = cj (l). Theorem Xj=1 Xl=1 Fix strategies for the k players s = (s ,..., s ). Every congestion game has a pure Nash equilibrium. 1 k What happens when Player i changes from si to si0? ∆costi = costi (si0, s i ) costi (si , s i ) − − − = c (n (s) + 1) c (n (s)) j j − j j j (s s ) j (s s ) ∈Xi0− i ∈Xi − i0 increased cost decreased cost

= Φ(si0, s i ) Φ(si , s i ) | − {z− −} | {z } = ∆Φ.

51 / 127 The potentialΦ reﬂects how a player’s costs change. 52 / 127 Congestion games Potential games

Proof If we start at an arbitrary s, and update one player’s choice to Definition decrease that player’s cost, the potential must decrease. Consider a multiplayer game G: Continuing updating other player’s strategies in this way, we must k players eventually reach a local minimum (there are only finitely many For player i, there is a set Si of strategies. strategies). For player i, there is costi : S1 Sk R. × · · · × → Since no player can reduce their cost from there, we have reached a We sayΦ: S1 Sk R is a potential for game G if ∆Φ = ∆costi , pure Nash equilibrium. × · · · × → that is, for all i, s and si0, Φ(si0, s i ) Φ(si , s i ) = costi (si0, s i ) costi (si , s i ). − − − − − − This gives an algorithm for finding a pure Nash equilibrium: Update We say that G is a potential game if it has a potential. the choice of one player at a time to improve their cost.

53 / 127 54 / 127 Potential games Topics 2

General-sum games, Nash equilibria. Two-player: payoﬀ matrices, dominant strategies, safety strategies. Multiplayer: Utility functions, Nash’s Theorem In considering congestion games, we proved two things: Congestion games and potential games Every potential game has a pure Nash equilibrium Theorem Every congestion game is a potential game 1 Every congestion game is a potential game. Other equilibrium concepts 2 Every potential game has a pure Nash equilibrium. Evolutionarily stable strategies Correlated equilibrium Price of anarchy Braess’s paradox The impact of adding edges Classes of latencies Pigou networks

55 / 127 56 / 127 Alternatives to Nash equilibria? Evolutionarily stable strategies

There is a population of individuals. Alternative equilibrium concepts There is a game played between pairs of individuals. 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.

57 / 127 58 / 127 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. x’s utility: x>A (z + (1 )x) = x>Az + (1 )x>Ax Suppose that it is symmetric (A = B>). − − Consider a mixed strategy x. z’s utility: z>A (z + (1 )x) = z>Az + (1 )z>Ax − − Think of x as the proportion of each pure strategy in the population.

Deﬁnition Mixed strategy x ∆ is an evolutionarily stable strategy (ESS) if, for any ∈ n pure strategy z, 1 z Ax x Ax (x, x) is a Nash equilibrium. > ≤ > ←− 2 If z>Ax = x>Ax then z>Az < x>Az.

59 / 127 60 / 127 ESS and Nash equilibria Topics 2

Theorem General-sum games, Nash equilibria. Every ESS is a Nash equilibrium. Two-player: payoff matrices, dominant strategies, safety strategies. Multiplayer: Utility functions, Nash’s Theorem Definition Congestion games and potential games Every potential game has a pure Nash equilibrium A strategy profile x = (x ,..., x ) ∆ ∆ is a strict Nash 1∗ k∗ ∈ S1 × · · · × Sk Every congestion game is a potential game equilibrium for utility functions u ,..., u if, for each player 1 k Other equilibrium concepts j 1,..., k , for all xj ∆S that is different from x∗, ∈ { } ∈ j j Evolutionarily stable strategies Correlated equilibrium uj (xj , x∗ j ) < uj (xj∗, x∗ j ). − − Price of anarchy Braess’s paradox The impact of adding edges Theorem Classes of latencies Every strict Nash equilibrium is an ESS. Pigou networks

61 / 127 62 / 127 Correlated equilibria: A driving example Correlated equilibrium

Payoff Definition Go Stop For a two player game with strategy sets S = 1,..., m and 1 { } Go (-100,-100) (1,-1) S = 1,..., n , a correlated strategy pair is a pair of random variables 2 { } Stop (-1,1) (-1,-1) (R, C) with some joint probability distribution over pairs of actions (i, j) S S . ∈ 1 × 2 Nash equilibria 2 99 2 99 Example (Go, Stop), (Stop, Go), 101 , 101 , 101 , 101 . (because we want indifference: 100p + 1 p = p (1 p)) In the traffic signal example, Pr(Stop, Go) = Pr(Go, Stop) = 1/2. − − − − − Better solution c.f. a pair of mixed strategies

A traffic signal: Pr((Red, Green)) = Pr((Green, Red)) = 1/2, and If we have x ∆Sm and y ∆Sn , then choosing the two actions( R, C) ∈ ∈ both players agree: Red means Stop, Green means Go. independently gives Pr(R = i, C = j) = xi yj . After they both see the traffic signal, the players have no incentive to In the traffic signal example, we cannot have Pr(Stop, Go) > 0 and deviate from the agreed actions. Pr(Go, Stop) > 0 without Pr(Go, Go) > 0.

63 / 127 64 / 127 Correlated equilibrium Topics 2

Definition General-sum games, Nash equilibria. A correlated strategy pair for a two-player game with payoff matrices A Two-player: payoff matrices, dominant strategies, safety strategies. and B is a correlated equilibrium if Multiplayer: Utility functions, Nash’s Theorem 1 for all i, i 0 S1, Pr(R = i) > 0 E [ai,C R = i] E ai ,C R = i . Congestion games and potential games ∈ ⇒ | ≥ 0 | 2 for all j, j S , Pr(C = j) > 0 [b C = j] b C = j . Every potential game has a pure Nash equilibrium 0 2 E R,j E R,j0 ∈ ⇒ | ≥ | Every congestion game is a potential game c.f. a Nash equilibrium Other equilibrium concepts Evolutionarily stable strategies A strategy profile( x, y) ∆ ∆ is a Nash equilibrium iff ∈ Sm × Sn Correlated equilibrium 1 for all i, i 0 S1, Pr(R = i) > 0 E [ai,C ] E ai ,C . Price of anarchy ∈ ⇒ ≥ 0 2 for all j, j0 S2, Pr(C = j) > 0 E [bR,j ] E bR,j . Braess’s paradox ∈ ⇒ ≥ 0 When R and C are independent, these expectations and the conditional The impact of adding edges Classes of latencies expectations are identical. Pigou networks Thus, a Nash equilibrium is a correlated equilibrium.

65 / 127 66 / 127 Braess’s paradox Price of anarchy

Example: before Deﬁnition For a routing problem, deﬁne average travel time in worst Nash equilibrium price of anarchy = . minimal average travel time

Example: after 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.

(Karlin and Peres, 2016)67 / 127 68 / 127 Price of anarchy Price of anarchy

Example: price of anarchy=4 /3

Example: Nash equals socially optimal

(Karlin and Peres, 2016) 69 / 127 70 / 127 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. We write the flow on an edge e as Theorem Fe = fP . 1 For linear latency functions, the price of anarchy is1. P e X3 2 For affine latency functions, the price of anarchy is no more than4 /3.

Latency on an edge e is a non-decreasing function of Fe , written `e (Fe ).

Latency on a path P is the total latency, LP (f ) = e P è (Fe ). ∈ Average latency is L(f ) = P fP LP (f ) = e Fe è (PFe ). A flow f is a Nash equilibriumP if, for all PPand P0, if fP > 0, L (f ) L (f ). P ≤ P0

71 / 127 72 / 127 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 = x ax : a 0 , L { 7→ ≥ } Consider a network G with a Nash equilibrium ﬂow fG and average latency = x ax + b : a, b 0 , L (f ), and a network H with additional roads added. Suppose that the L { 7→ ≥ } G G What about price of anarchy in H is no more than α. Then any Nash equilibrium ﬂow d fH has average latency LH (fH ) αLG (fG ). = x ad x : ad 0 ? ≤ L ( 7→ ≥ ) Xd 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 is at most the price of anarchy in a L certain small network with these latency functions: a Pigou network.

73 / 127 74 / 127 Pigou networks and price of anarchy 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:

r`(r) αr (`) = . minx 0(x`(x) + (r x)`(r)) ≥ − For any network with latency functions from and total ﬂow1, the price L of anarchy is no more than

max max αr (`). 0 r 1 ` ≤ ≤ ∈L

(Karlin and Peres, 2016)

75 / 127 76 / 127 Topics 2 Topics 3

Cooperative games General-sum games, Nash equilibria. Transferable versus nontransferable utility Two-player: payoff matrices, dominant strategies, safety strategies. Two-player transferable utility cooperative games Multiplayer: Utility functions, Nash’s Theorem Cooperative strategy, threat strategies, disagreement point, final payoff Congestion games and potential games vector Two-player nontransferable utility cooperative games Every potential game has a pure Nash equilibrium Bargaining problems Every congestion game is a potential game Nash’s bargaining axioms, the Nash bargaining solution Other equilibrium concepts Multi-player transferable utility cooperative games Evolutionarily stable strategies Characteristic function Correlated equilibrium Gillies’ core Price of anarchy Shapley’s axioms, Shapley’s Theorem Braess’s paradox Designing games The impact of adding edges Voting systems. Classes of latencies Voting rules, ranking rules Pigou networks Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem Properties of voting rules Instant runoff voting, Borda count, positional voting rules

77 / 127 78 / 127 Cooperative versus noncooperative games Cooperative versus noncooperative games

Cooperative games Noncooperative games Players can make binding agreements. Players play their strategies simultaneously. e.g.: prisoner’s dilemma. They might communicate (or see a common signal; e.g., a traﬃc Both players gain from an enforceable agreement not to confess. signal), but there’s no enforced agreement. Two types: Natural solution concepts: Transferable utility The players agree what strategies to play and Nash equilibrium, correlated equilibrium. what additional side payments are to be made. No improvement from unilaterally deviating. Nontransferable utility The players choose a joint strategy, but there are no side payments.

79 / 127 80 / 127 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. (3,6) A feasible payoff vector( v1, v2) is Pareto optimal if the only feasible Payoffs (5,5) payoff vector( v 0 , v 0 ) with v 0 v1 and v 0 v2 is( v 0 , v 0 ) = (v1, v2). 1 2 1 ≥ 2 ≥ 1 2 1 2 3 With transferable utility, the players can choose any payoff vector in 1 (2,2) (6,2) (1,2) the convex hull of the set of lines 2 (4,3) (3,6) (5,5) (4,3)

(aij p, bij + p): i 1,..., m , j 1,..., n , p R . { − ∈ { } ∈ { } ∈ } (1,2) (2,2) (6,2)

81 / 127 82 / 127 TU cooperative games TU cooperative games

Negotiation Solving two-player TU games

Players negotiate a joint strategy and a side payment. 1 Find the cooperative strategy( i0, j0) with

Since they are rational, they will agree to play a Pareto optimal payoff ai0j0 + bi0j0 = maxi,j (aij + bij ). vector. 2 For the zero-sum game with payoff matrix A B, find the optimal − Players might make threats (and counter-threats) to justify their strategies x , y (threat strategies) and the value δ = x>(A B)y . ∗ ∗ ∗ − ∗ desired payoff vectors. 3 The disagreement point is( d1, d2) = x>Ay , x>By ∗ ∗ ∗ ∗ If agreement is not reached, they could carry out their threats. 4 The final payoff vector is (a , b ) = ((σ d + d )/2, (σ d + d )/2), with σ = a + b . But reaching agreement gives higher utility, so the threats are only ∗ ∗ − 2 1 − 1 2 i0j0 i0j0 relevant to choosing a reasonable side payment. 5 The payment from Player I to Player II is a a . i0j0 − ∗

83 / 127 84 / 127 TU cooperative games Topics 3

Key ideas Cooperative games Transferable versus nontransferable utility Cooperative strategy: biggest entry in A + B. Two-player transferable utility cooperative games Disagreement point: solution to zero-sum game with payoff matrix Cooperative strategy, threat strategies, disagreement point, final payoff A B. vector − Final payoff vector: midpoint of the Pareto optimal region. Two-player nontransferable utility cooperative games Bargaining problems Nash’s bargaining axioms, the Nash bargaining solution Multi-player transferable utility cooperative games Characteristic function (3,6) Gillies’ core (5,5) Shapley’s axioms, Shapley’s Theorem Designing games (4,3) Voting systems. (1,2) (2,2) (6,2) Voting rules, ranking rules Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem Properties of voting rules Instant runoff voting, Borda count, positional voting rules

85 / 127 86 / 127 Nontransferable utility cooperative games Nash Bargaining Model for NTU games

Feasible payoffs (NTU) Ingredients of a bargaining problem 1 A compact, convex feasible set S R2. ⊂ 2 (3,6) 2 A disagreement point d = (d1, d2) R . ∈ Payoffs (5,5) Think of the disagreement point as the utility that the players get from walking away and not playing the game. 1 2 3 (And we’ll assume every x S has x d , x d , with strict 1 (2,2) (6,2) (1,2) ∈ 1 ≥ 1 2 ≥ 2 (4,3) inequalities for some x S.) 2 (4,3) (3,6) (5,5) ∈ (1,2) (2,2) (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 = (a , a ) S. 1 2 ∈

87 / 127 88 / 127 Nash Bargaining Model for NTU games Nash Bargaining Model for NTU games

Nash’s bargaining axioms Nash’s bargaining axioms 1 Pareto optimality: the only feasible payoff vector( v , v ) with v a Pareto optimality: The agreement point shouldn’t be dominated by 1 2 1 ≥ 1 and v a is( v , v ) = (a , a ). another point for both players. (Criticism: why should one player care 2 ≥ 2 1 2 1 2 2 Symmetry: If both( x, y) S implies( y, x) S and d = d then if the agreement point is only dominated for the other player?) ∈ ∈ 1 2 a1 = a2. Symmetry: This is about fairness: if nothing distinguishes the players, 3 Affine covariance: For any affine transformation the solution should be similarly symmetric. Ψ(x1, x2) = (α1x1 + β1, α2x2 + β2) with α1, α2 > 0 and any S and d, Affine covariance: Changing the units (or a constant offset) of the F (Ψ(S), Ψ(d)) = Ψ(F (S, d)). utilities should not affect the outcome of bargaining. 4 Independence of irrelevant attributes: For two bargaining problems Independence of irrelevant attributes: This assumes that all of the (R, d) and( S, d), if R S and F (S, d) R, then threats the players might make have been accounted for in the ⊂ ∈ F (R, d) = F (S, d). disagreement point.

89 / 127 90 / 127 Nash Bargaining Model for NTU games Topics 3

Cooperative games Transferable versus nontransferable utility Theorem Two-player transferable utility cooperative games There is a unique function F satisfying Nash’s bargaining axioms. Cooperative strategy, threat strategies, disagreement point, final payoff It is the function that takes S and d and returns the unique solution to the vector optimization problem Two-player nontransferable utility cooperative games Bargaining problems Nash’s bargaining axioms, the Nash bargaining solution max (x1 d1)(x2 d2) (x1,x2) − − Multi-player transferable utility cooperative games subject to x d Characteristic function 1 ≥ 1 Gillies’ core x d Shapley’s axioms, Shapley’s Theorem 2 ≥ 2 (x1, x2) S. 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

91 / 127 92 / 127 Multiplayer TU cooperative games The core

Characteristic function We focus on the formation of coalitions in multiplayer TU games. Gillies’ core Define a characteristic function: for each subset S of players, v(S) is Efficiency n ψ (v) = v( 1,..., n ). i=1 i { } the total value that would be available to be split by that subset of Stability ForP all S 1,..., n , i S ψi (v) v(S). players, no matter what the other players do. ⊆ { } ∈ ≥ P (Called a coalitional form game, versus a strategic form game.) These seem reasonable properties: 1 The total payoff gets allocated. Anything less is irrational. Allocations 2 A coalition gets allocated at least the payoff it can obtain on its own. Define an allocation function ψ as a map from a characteristic function v But there might be no allocation, one allocation, or many allocations in for n players to a vector ψ(v) Rn. the core! ∈ This is the payoff that is allocated to the n players.

93 / 127 94 / 127 Shapley value Shapley value

Shapley’s Theorem Shapley axioms The following allocation uniquely satisfies Shapley’s axioms: Efficiency n ψ (v) = v( 1,..., n ). i=1 i { } ψi (v) = Eπφi (v, π), Symmetry If, for all S 1,..., n not containing i, j, P ⊂ { } v(S i ) = v(S j ), then ψi (v) = ψj (v). where the expectation is over uniformly chosen permutations π on ∪ { } ∪ { } 1,..., n , and Dummy If for all S 1,..., n , v(S i ) = v(S), then ψ (v) = 0. i { } 1 1 ⊂ { } ∪ { } φi (v, π) = v π 1, . . . , π− (i) v π 1, . . . , π− (i) 1 . Additivity ψi (v + u) = ψi (v) + ψi (u). { } − { − } Shapley’s Theorem Example: For the identity permutation, π(i) = i, Shapley’s axioms uniquely determine the allocation ψ. φ (v, π) = v ( 1,..., i ) v ( 1,..., i 1 ) , i { } − { − } which is how much value i adds to 1,..., i 1 . We call the unique allocation ψ(v) the Shapley value of the players in the { − } game defined by the characteristic function v. And for a random π, φi (v, π) is how much value i adds to the random set π 1, . . . , π 1(i) 1 . { − − } 95 / 127 96 / 127 Shapley value Topics 3

Example: Junta game (J-veto game) Cooperative games Transferable versus nontransferable utility Two-player transferable utility cooperative games 1 if J S, 1 Cooperative strategy, threat strategies, disagreement point, final payoff wJ (S) = 1[J S] = ⊆ ψi (wJ ) = 1 [i J] . ⊆ (0 otherwise. ∈ J vector | | Two-player nontransferable utility cooperative games Bargaining problems Lemma [Characteristic functions as junta games] Nash’s bargaining axioms, the Nash bargaining solution Multi-player transferable utility cooperative games We can write any v as a unique linear combination of wJ ’s. Characteristic function Gillies’ core Computing Shapley value Shapley’s axioms, Shapley’s Theorem We can always write the characteristic function as Designing games Voting systems. v(S) = J cJ wJ (S), where the sum is over nonempty subsets J 1,..., n . Voting rules, ranking rules ⊆ { P } Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem We know that ψ (w ) = 1[i J]/ J . Properties of voting rules i J ∈ | | This is often an easy approach to computing the Shapley value. Instant runoff voting, Borda count, positional voting rules 97 / 127 98 / 127 Designing games and mechanisms Social choice and Voting

We aim to design the rules of a game so that the outcomes have certain Electing a president desired properties. Suppose there are two candidates for president, and all voters have a Elections Consistent with voters’ rankings preference. Fair (symmetric) How do we design an election to decide between the two candidates? Auctions Maximize revenue for the seller Voters vote; the candidate with the most votes wins. Pareto eﬃciency The candidate that wins is the choice of at least half of the voters. Calibrated (revealing bidders’ values) Voters never have an incentive to vote against their preferences. Tournaments The best team is most likely to win Things aren’t as simple with three candidates. Players have an incentive to compete

99 / 127 100 / 127 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 Questions vector 1 How do we model voters’ preferences? Two-player nontransferable utility cooperative games 2 Bargaining problems How do voters’ express their preferences? Nash’s bargaining axioms, the Nash bargaining solution 3 How do we combine that information? Multi-player transferable utility cooperative games Characteristic function We will distinguish two outcomes: Gillies’ core Shapley’s axioms, Shapley’s Theorem 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

101 / 127 102 / 127 Topics 3 Voting and Ranking

Cooperative games Transferable versus nontransferable utility Assumptions Two-player transferable utility cooperative games Cooperative strategy, threat strategies, disagreement point, final payoff There is a setΓ of candidates. vector Voter i has a preference relation “ ” defined on candidates that is: i Two-player nontransferable utility cooperative games 1 Complete: A = B Γ, A B or B A. ∀ 6 ∈ i i Bargaining problems 2 Transitive: A, B, C Γ, A i B and B i C implies A i C. Nash’s bargaining axioms, the Nash bargaining solution ∀ ∈ Multi-player transferable utility cooperative games Definitions Characteristic function Gillies’ core A voting rule f maps a preference profile π = ( ,..., ) to a 1 n Shapley’s axioms, Shapley’s Theorem winner fromΓ. Designing games A ranking rule R maps a preference profile π = ( ,..., ) to a 1 n Voting systems. social ranking “ ” onΓ, which is another complete, transitive Voting rules, ranking rules preference relation. Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem Properties of voting rules Instant runoff voting, Borda count, positional voting rules

103 / 127 104 / 127 Properties of ranking rules Properties of ranking rules

Strategically vulnerable A ranking rule R is strategically This means that Voter i has Unanimity vulnerable if, for some preference a preference relation i , but A ranking rule R has the If all voters prefer candidate proﬁle( ,..., ), some voter by stating an alternative 1 n unanimity property if, for all i, A over B, candidate A i and some candidates A, B Γ, preference relation 0, it can i A B, then = R( ,..., ) should be ranked above B. ∈ swap the ranking rule’s i 1 n satisﬁes A B. := R( 1,..., i ,..., n), preference between A and B 0 := R( 1,..., 0,..., n), to make it consistent with i i . A B, B A, but A B. i 0

105 / 127 106 / 127 Properties of ranking rules Properties of ranking rules

Theorem Any ranking rule R that violates IIA is strategically vulnerable. Independence of irrelevant alternatives (IIA) Definition Consider two different voter The ranking rule’s relative A ranking rule R is a dictatorship if there is a voter i ∗ such that, for any preference profiles( ,..., ) rankings of candidates A and preference profile( 1,..., n), = R( 1,..., n) has A B iff A i ∗ B. 1 n and( ,..., ), and define B should depend only on the 10 n0 = R( ,..., ) and voters’ relative rankings of Arrow’s Impossibility Theorem 1 n 0= R( 10 ,..., n0 ). For these two candidates. For Γ 3, any ranking rule R that satisfies both IIA and unanimity is a | | ≥ A, B Γ, if, for all i, A i B iff ∈ dictatorship. A B, then A B iff A B. i0 0 Thus, any ranking rule R that satisfies unanimity and is not strategically vulnerable is a dictatorship.

Hence, strategic vulnerability is inevitable.

107 / 127 108 / 127 Topics 3 Strategic vulnerability

Cooperative games Transferable versus nontransferable utility Two-player transferable utility cooperative games Recall: Ranking Voting Cooperative strategy, threat strategies, disagreement point, final payoff A ranking rule R is strategically A voting rule f is strategically vector vulnerable if, for some preference vulnerable if, for some preference Two-player nontransferable utility cooperative games profile( ,..., ), some voter profile( 1,..., n), some voter i 1 n Bargaining problems i and some candidates A, B Γ, and some candidates A, B Γ, Nash’s bargaining axioms, the Nash bargaining solution ∈ ∈ Multi-player transferable utility cooperative games := R( 1,..., i ,..., n), π := ( 1,..., i ,..., n), Characteristic function 0 := R( ,..., 0,..., ), π0 := ( 1,..., 0,..., n), Gillies’ core 1 i n i Shapley’s axioms, Shapley’s Theorem A B, B A, but A B. A i B, B = f (π), but A = f (π0). Designing games i 0 Voting systems. Voter i, by incorrectly reporting preferences, can change the outcome Voting rules, ranking rules to match his true preferences. Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem Properties of voting rules Instant runoff voting, Borda count, positional voting rules

109 / 127 110 / 127 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 A voting rule f is a dictatorship if A voting rule f maps from the voters’ preference profile π to a winner if there is a voter i such that, there is a voter i such that, for ∗ ∗ inΓ. for any preference profile any preference profile We say f is onto Γ if, for all candidates A Γ, there is a π satisfying ( 1,..., n), = R( 1,..., n) ( 1,..., n), A = f ( 1,..., n) ∈ f (π) = A. has A B iff A i B. iff for all B = A, A i B. ∗ 6 ∗ If f is not ontoΓ, some candidate is excluded from winning. Voter i ∗ determines the outcome. Gibbard-Satterthwaite Theorem For Γ 3, any voting rule f that is ontoΓ and is not strategically | | ≥ vulnerable is a dictatorship.

111 / 127 112 / 127 Topics 3 Properties of voting systems

Cooperative games Transferable versus nontransferable utility Symmetry: Permuting voters does not affect the outcome. Two-player transferable utility cooperative games Monotonicity: Changing one voter’s preferences by promoting Cooperative strategy, threat strategies, disagreement point, final payoff candidate A without changing any other preferences should not vector Two-player nontransferable utility cooperative games change the outcome from A winning to A not winning. Bargaining problems Condorcet winner criterion: If a candidate is majority-preferred in Nash’s bargaining axioms, the Nash bargaining solution pairwise comparisons with all other candidates, then that candidate Multi-player transferable utility cooperative games wins. Characteristic function Gillies’ core Condorcet loser criterion: If a candidate is preferred by a minority Shapley’s axioms, Shapley’s Theorem of voters in pairwise comparisons with all other candidates, then that Designing games candidate should not win. Voting systems. Smith criterion The winner always comes from the Smith set (the Voting rules, ranking rules smallest nonempty set of candidates that are majority-preferred in Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem pairwise comparisons with any candidate outside the set). Properties of voting rules Instant runoff voting, Borda count, positional voting rules

113 / 127 114 / 127 Properties of voting systems Topics 3

Cooperative games Reversal symmetry: If A wins for some voter preference profile, A Transferable versus nontransferable utility Two-player transferable utility cooperative games does not win when the preferences of all voters are reversed. Cooperative strategy, threat strategies, disagreement point, final payoff Cancellation of ranking cycles: If a set of Γ voters have vector | | preferences that are cyclic shifts of each other (e.g., A B C, Two-player nontransferable utility cooperative games 1 1 B C A, C A B), then removing these voters does not Bargaining problems 2 2 3 3 affect the outcome. Nash’s bargaining axioms, the Nash bargaining solution Multi-player transferable utility cooperative games Cancellation of opposing rankings: If two voters have reversed Characteristic function preferences, then removing these voters does not affect the outcome. Gillies’ core Shapley’s axioms, Shapley’s Theorem Consistency: If A wins for voter preference profiles π and π0, A also wins when these voter preference profiles are combined. Designing games Voting systems. Participation: If A wins for some voter preference profile, then Voting rules, ranking rules adding a voter with A B does not change the winner from A to B. Arrow’s impossibility theorem, Gibbard-Satterthwaite Theorem Properties of voting rules Instant runoff voting, Borda count, positional voting rules

115 / 127 116 / 127 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. Properties of instant runoff voting Otherwise: 1 Eliminate the candidate that is top-ranked by the fewest voters. Monotonicity? (Changing one voter’s preferences by promoting 2 Drop that candidate’s preferences from voters’ rankings. candidate A without changing any other preferences does not change 3 Use instant runoff voting on the remaining candidates with the the outcome from A winning to A not winning.) No. reassigned preferences.

Properties of instant runoﬀ voting Symmetry? (Permuting voters does not aﬀect the outcome.) Yes.

117 / 127 118 / 127 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 Condorcet loser criterion? (If a candidate is preferred by a minority pairwise comparisons with all other candidates, then that candidate of voters in pairwise comparisons with all other candidates, then that wins.) No. candidate should not win.) Yes.

119 / 127 120 / 127 Properties of voting systems Properties of voting systems

Properties of instant runoff voting Properties of instant runoff voting Reversal symmetry? (If A wins for some voter preference profile, A Smith criterion? (The winner always comes from the Smith set—the does not win when the preferences of all voters are reversed.) No. smallest nonempty set of candidates that are majority-preferred in Cancellation of ranking cycles? No. pairwise comparisons with any candidate outside the set.) No. Cancellation of opposing rankings? No. (Notice that a rule that violates the Condorcet winner criterion violates Consistency? No. the Smith criterion for some preference profile with a singleton Smith set.) Participation? No.

121 / 127 122 / 127 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 Two-player nontransferable utility cooperative games Borda count Bargaining problems Voters rank candidates from1 to N (where N = Γ ). Nash’s bargaining axioms, the Nash bargaining solution | | A candidate that is ranked in ith position is assigned N i + 1 points. Multi-player transferable utility cooperative games − Characteristic function The candidate with the largest total wins. 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

123 / 127 124 / 127 Properties of voting systems Properties of voting systems

Positional voting rules Properties of positional voting rules Deﬁne a a a . 1 ≥ 2 ≥ · · · ≥ N Symmetry? Yes. For each candidate, assign ai points for each voter that assigns that Monotonicity? Yes. candidate rank i. Condorcet winner criterion? No. The candidate with the largest total wins. Cancellation of ranking cycles? Yes. e.g., Borda count: N, N 1,..., 1. Consistency? Yes. − e.g., Plurality:1 , 0,..., 0. Participation? Yes. e.g., Approval voting:1 , 1,..., 1, 0,..., 0.

125 / 127 126 / 127 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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