Economics and Computer Science Introduction to Game Theory CS595, SB 213 Xiang-Yang Li Department of Computer Science Illinois Institute of Technology

Selfish Users How to model this? Turn to game theory How to achieve a global system gold when selfish users are present? Economics results How to implement this? Combine with cryptography and security Is it efficient? Combine with traditional computer science wisdoms

The game view of business Five basic elements of a game (PARTS) Players Added values Rules Tactics Scope

Value Net customers competitors Company complementors suppliers

Value Net Complementor Competitor A player is your complementor if customers value your product more when they have the other players product than they have your product alone. Inter vs. Microsoft Competitor A player is your complementor if customers value your product less when they have the other players product than they have your product alone. Coca-cola vs. Pepsi-cola

Game Theory An example: Prof. Adam and 26 students Adam keeps 26 black cards and distributes 26 red cards one to each student Dean offer $100 for a pair of red and black cards Restriction: students cannot gather together and bargain as a group with Adam. What will each negotiation end up? 50/50 split

What happens if Another example (Barry’s card game): Prof. Adam and 26 students Adam keeps 23 black cards and distributes 26 red cards one to each student Dean offer $100 for a pair of red and black cards Restriction: students cannot gather together and bargain as a group with Adam. What will each negotiation end up? Likely 90/10 split

Added Value Your added value= Example Size of the pie when you are in the game minus the size of the pie when you are out of the game Example Card game one Added value of Adam is $2600, each student is $100, so total added value is $5200 Barry’s game Added value of Adam is $2300, each student is $0, so total added value is $2300!

What does it tell? Instead of focusing on the minimum payoff you are willing to accept, be sure to consider how much the other players are willing to pay to have you in the game! Do not confuse your individual added value with the larger added value of a group of people in the same position of the game as you Example: Barry’s card game

Rules Rules can change the game Card game example: Rule: take-it-or-leave-it negotiation: a student can either accept or reject the offer by Adam, but not counter-offer, nor second offer from Adam. What will the negotiation turn out to be? A 50/50 split or 90/10 split or something else Who is more powerful now?

Rationality and Irrationality Game theory assumes rational player Maximize its profits Understand the game No misperceptions No feelings of pride No fairness No jealousy, spite, vengefulness, altruism But the world is not like this So much for game theory, 

What is rationality Rationality means A player is rational if he does the best he can, given how he perceives the game, including his perceptions of perceptions, and how he evaluates the various possible outcomes of the game A player can percept wrong and still be rational: he is doing the best he can given what he knows.

Nash Equilibrium: best choice if others fixed Prisoner’s Dilemma Bob Confess Decline +5 , +5 0 ,+10 +10 ,0 +2 ,+2 Alice Dominant strategy: best choice no matter others do Nash Equilibrium: best choice if others fixed

What is game theory?

Types of game Strategic games Extensive Games A strategic game is a model of interactive decision-making in which each decision-maker chooses his plan of action once and for all, and these choices are made simultaneously. Extensive Games An extensive game is a detailed description of the sequential structure of the decision problems encountered by the players in a strategic situation. Perfect information/imperfect information: whether each player when making any decision is perfectly informed of all the events that have previously occurred or not.

Strategic games It consists of A finite set N of players For each player , a nonempty set of actions available to player i. For each player i, a preference relation on all the actions taken by all players Game is called finite if the set of actions of each player is finite.

Utility function Under a wide range of circumstances the preference relation of player I is represented by a payoff function:

Agenthood Agent attempts to maximize its expected utility We use economic definition of agent as locus of self-interest Could be implemented e.g. as several mobile “agents” … Agent attempts to maximize its expected utility Utility function ui of agent i is a mapping from outcomes to reals Can be over a multi-dimensional outcome space Incorporates agent’s risk attitude (allows quantitative tradeoffs) E.g. outcomes over money

Agenthood u 1 0.5 M$ Lottery 1: $0.5M w.p. 1 Lottery 2: $1M w.p. 0.5 i Risk seeking Risk neutral Risk averse 1 0.5 Lottery 1: $0.5M w.p. 1 Lottery 2: $1M w.p. 0.5 $0 w.p. 0.5 Agent’s strategy is the choice of lottery Risk aversion => insurance companies

Utility functions are scale-invariant Agent i chooses a strategy that maximizes expected utility maxstrategy Soutcome p(outcome | strategy) ui(outcome) If ui’() = a ui() + b for a > 0 then the agent will choose the same strategy under utility function ui’ as it would under ui Note that ui has to be finite for each possible outcome Otherwise expected utility could be infinite for several strategies, so the strategies could not be compared.

Full vs bounded rationality Full rationality Bounded rationality Descriptive vs. prescriptive theories of bounded rationality

Criteria for evaluating multiagent systems Computational efficiency Distribution of computation Communication efficiency Social welfare: maxoutcome ∑i ui(outcome) Requires cardinal utility comparison … but we just said that utility functions are arbitrary in terms of scale! Surplus: social welfare of outcome – social welfare of status quo Constant sum games have 0 surplus. Markets are not constant sum Pareto efficiency: An outcome o is Pareto efficient if there exists no other outcome o’ s.t. some agent has higher utility in o’ than in o and no agent has lower Implied by social welfare maximization Individual rationality: Participating in the negotiation (or individual deal) is no worse than not participating Stability: No agents can increase their utility by changing their strategies Symmetry: No agent should be inherently preferred, e.g. dictator

Terminology In a 1-agent setting, agent’s expected utility maximizing strategy is well-defined But in a multiagent system, the outcome may depend on others’ strategies also Game theory analyzes stable points in the space of strategy profiles => allows one to build robust, nonmanipulable multiagent systems Agent = player Action = move = choice that agent can make at a point in the game Strategy si = mapping from history (to the extent that the agent i can distinguish) to actions Strategy set Si = strategies available to the agent Strategy profile (s1, s2, ..., s|A|) = one strategy for each agent Agent’s utility is determined after each agent (including nature that is used to model uncertainty) has chosen its strategy, and game has been played: ui = ui(s1, s2, ..., s|A|)

Game representations Extensive form Matrix form (aka normal form player 1 1, 2 3, 4 player 2 Up Down Left Right 5, 6 7, 8 Matrix form (aka normal form aka strategic form) player 1’s strategy player 2’s strategy 1, 2 Up Down Left, Left Right 3, 4 5, 6 7, 8 Right, Potential combinatorial explosion

Nash Equilibrium A steady state of the play of a strategic game in which each player holds the correct expectation about the other player’s behavior and acts rationally It does NOT attempt to examine the process by which a steady state is reached.

Nash equilibrium [Nash50] Sometimes an agent’s best response depends on others’ strategies: a dominant strategy does not exist A strategy profile is a Nash equilibrium if no player has incentive to deviate from his strategy given that others do not deviate: for every agent i, ui(si*,s-i) ≥ ui(si’,s-i) for all si’ Dominant strategy equilibria are Nash equilibria but not vice versa Defect-defect is the only Nash eq. in Prisoner’s Dilemma Battle of the Sexes (has no dominant strategy equilibria):

Criticisms of Nash equilibrium Not unique in all games, e.g. Battle of the Sexes Approaches for addressing this problem Refinements of the equilibrium concept Eliminate weakly dominated strategies first Choose the Nash equilibrium with highest welfare Subgame perfection … Focal points Mediation Communication Convention Learning Does not exist in all games May be hard to compute Finding a good one is NP-hard [Gilboa&Zemel GEB-89], [Conitzer&Sandholm IJCAI-03]

Existence of (pure strategy) Nash equilibria Theorem. Any finite game, where each action node is alone in its information set (i.e. at every point in the game, the agent whose turn it is to move knows what moves have been played so far) is dominance solvable by backward induction (at least as long as ties are ruled out) Constructive proof: Multi-player minimax search

Rock-scissors-paper game Sequential moves

Rock-scissors-paper game Simultaneous moves

Mixed strategy Nash equilibrium Mixed strategy = agent’s chosen probability distribution over pure strategies from its strategy set Each agent has a best response strategy and beliefs (consistent with each other) move of agent 1 agent 2 rock scissors paper 0, 0 1, -1 -1, 1 Symmetric mixed strategy Nash eq: Each player plays each pure strategy with probability 1/3 In mixed strategy equilibrium, each strategy that occurs in the mix of agent i has equal expected utility to i Information set (the mover does not know which node of the set she is in)

Existence mixed strategy Nash equilibria Every finite player, finite strategy game has at least one Nash equilibrium if we admit mixed strategy equilibria as well as pure [Nash 50] (Proof is based on Kakutani’s fix point theorem)

(for distributional bargaining) Ultimatum game (for distributional bargaining)

Subgame perfect equilibrium & credible threats Proper subgame = subtree (of the game tree) whose root is alone in its information set Subgame perfect equilibrium = strategy profile that is in Nash equilibrium in every proper subgame (including the root), whether or not that subgame is reached along the equilibrium path of play

Subgame perfect equilibrium & credible threats E.g. Cuban missile crisis Pure strategy Nash equilibria: (Arm,Fold), (Retract,Nuke) Pure strategy subgame perfect equilibria: (Arm,Fold) Conclusion: Kennedy’s Nuke threat was not credible Khrushchev Kennedy Arm Retract Fold Nuke -1, 1 - 100, - 100 10, -10

Thoughts on credible threats Could use software as a commitment device If one can credibly convince others that one cannot change one’s software agent, then revealing the agent’s code acts as a credible commitment to one’s strategy E.g. nuke in the missile crisis E.g. accept no less than 60% as the second mover in the ultimatum game Restricting one’s strategy set can increase one’s utility This cannot occur in single agent settings Social welfare can increase or decrease

Conclusions on game-theoretic tools Tools for building robust, nonmanipulable systems with self-interested agents and different agent designers Different solution concepts For existence, use strongest equilibrium concept For uniqueness, use weakest equilibrium concept

Implementation in dominant strategies Strongest form of mechanism design

Dominant strategy equilibrium Best response si*: for all si’, ui(si*,s-i) ≥ ui(si’,s-i) Dominant strategy si*: si* is a best response for all s-i Does not always exist Inferior strategies are called dominated Dominant strategy equilibrium is a strategy profile where each agent has picked its dominant strategy Requires no counterspeculation cooperate defect 3, 3 0, 5 5, 0 1, 1 Pareto optimal? Social welfare maximizing?

Implementation in dominant strategies Goal is to design the rules of the game (aka mechanism) so that in dominant strategy equilibrium (s1, …, s|A|), the outcome of the game is f(u1, …, u|A|) Nice in that agents cannot benefit from counterspeculating each other Others’ preferences Others’ rationality Other’s endowments Other’s capabilities …

Gibbard-Satterthwaite impossibility Thrm. If |O | ≥ 3 (and each outcome would be the social choice under f for some input profile (u1, …, u|A|) ) and f is implementable in dominant strategies, then f is dictatorial Proof. (Assume for simplicity that utility relations are strict) By the revelation principle, if f is implementable in dominant strategies, it is truthfully implementable in dominant strategies with a direct revelation mechanism (maybe not in unique equilibrium) Since f is truthfully implementable in dominant strategies, the following holds for each agent i: ui(f(ui,u-i)) ≥ ui(f(ui’,u-i)) for all u-i Claim: f is monotonic. Suppose not. Then there exists u and u’ s.t. f(u) = x, x maintains position going from u to u’, and f(u’)  x Consider converting u to u’ one agent at a time. The social choices in this sequence are e.g. x, x, y, z, x, z, y, …, z. Consider the first step in this sequence where the social choice changes. Call the agent that changed his preferences agent i, and call the new social choice y. For the mechanism to be truth-dominant, i’s dominant strategy should be to tell the truth no matter what others reveal. So, truth telling should be dominant even if the rest of the sequence did not occur. Case 1. u’i(x) > u’i(y). Say that u’i is the agent’s truthful preference. Agent i would do better by revealing ui intead (x would get chosen instead of y). This contradicts truth-dominance. Case 2. u’i(x) < u’i(y). Because x maintains position from ui to u’i, we have ui(x) < ui(y). Say that ui is the agent’s truthful preference. Agent i would do better by revealing u’i instead (y would get chosen instead of x). This contradicts truth-dominance. Claim: f is Paretian. Suppose not. Then for some preference profile u we have an outcome x such that for each agent i, ui(x) > ui(f(u)). We also know that there exists a u’ s.t. f(u’) = x Now, choose a u’’ s.t. for all i, ui’’(x) > ui’’(f(u)) > ui’’(z), z  f(u), x Since f(u’) = x, monotonicity implies f(u’’) = x (because going from u’ to u’’, x maintains its position) Monotonicity also implies f(u’’) = f(u) (because going from u to u’’, f(u) maintains its position) But f(u’’) = x and f(u’’) = f(u) yields a contradiction because x  f(u) Since f is monotonic & Paretian, by strong form of Arrow’s theorem, f is dictatorial. QED

Ways around the Gibbard-Satterthwaite impossibility Use a weaker equilibrium notion In practice, agent might not know others’ revelations Design mechanisms where computing a beneficial manipulation (insincere ranking of outcomes) is hard NP-complete in second order Copeland voting mechanism [Bartholdi, Tovey, Trick 1989] Copeland score: Number of competitors an outcome beats in pairwise competitions 2nd order Copeland: Copeland, and break ties based on the sum of the Copeland scores of the competitors that the outcome beat NP-complete in Single Transferable Vote mechanism [Bartholdi & Orlin 1991] NP-hard, #P-hard, or PSPACE-hard in many voting protocols if one round of pairwise elimination is used before running the protocol [Conitzer & Sandholm IJCAI-03] Weighted coalitional manipulation (and thus unweighted individual manipulation when the manipulator has correlated uncertainty about others) is NP-complete in many voting protocols, even for a constant #candidates [Conitzer & Sandholm AAAI-02, Conitzer, Lang & Sandholm TARK-03] Randomization Agents’ preferences have special structure Need almost this much randomness General preferences Quasilinear preferences

Quasilinear preferences: VCG Outcome (x1, x2, ..., xk, m1, m2, ..., m|A| ) Quasilinear preferences: ui(x, m) = mi + vi(x1, x2, ..., xk) Utilitarian setting: Social welfare maximizing choice Outcome s(v1, v2, ..., v|A|) = maxx i vi(x1, x2, ..., xk) Thrm. Assume every agent’s utility function is quasilinear. A utilitarian social choice function f: v -> (s(v), m(v)) can be implemented in dominant strategies if mi(v)= ji vj(s(v)) + hi(v-i) for arbitrary function h Proof. We show that every agent’s (weakly) dominant strategy is to reveal the truth in this direct revelation (Groves) mechanism Let v be agents’ revealed preferences where agent i tells the truth Let v’ be the same revealed preferences, except that i lies Suppose agent i benefits from the lie: vi(s(v’)) + mi(v’) > vi(s(v)) + mi(v) That is, vi(s(v’)) + ji vj(s(v’)) + h i(v-i’) > vi(s(v)) + ji vj(s(v)) + h i(v-i) Because v-i’ = v-i we have h i(v-i’) = h i(v-i) Thus we must have vi(s(v’)) + ji vj(s(v’)) > vi(s(v)) + ji vj(s(v)) We can rewrite this as j vj(s(v’)) > j vj(s(v)) But this contradicts the definition of s() QED

Uniqueness of VCG Thrm. Assume every agent’s utility function is quasilinear. A utilitarian social choice function f: v -> (s(v), m(v)) can be implemented in dominant strategies for all v: A x O -> R only if mi(v)= ji vj(s(v)) + hi(v-i) for some function h Proof. Can write mi(v) = ji vj(s(v)) + hi(vi , v-i) We prove hi(vi , v-i) = hi(v-i) Suppose not, i.e., hi(vi , v-i)  hi(v’i , v-i) Case 1. s(vi , v-i) = s(v’i , v-i). If f is truthfully implementable in dominant strategies, we have that vi(s(vi , v-i)) + mi(vi , v-i)  vi(s(v’i , v-i)) + mi(v’i , v-i) and that v’i(s(v’i , v-i)) + mi(v’i , v-i)  v’i(s(vi , v-i)) + mi(vi , v-i) Since s(vi , v-i) = s(v’i , v-i), these inequalities imply hi(vi , v-i) = hi(v’i , v-i). Contradiction

Uniqueness of VCG PROOF CONTINUES… Case 2. s(vi , v-i)  s(v’i , v-i). Suppose wlog that hi(vi , v-i) > hi(v’i , v-i) Consider an agent with the following preference Let v’’i(x) = - ji vj(s(vi , v-i)) if x = s(vi , v-i) Let v’’i(x) = - ji vj(s(v’i , v-i)) +  if x = s(v’i , v-i) Let v’’i(x) = - otherwise We will show that v’’i will prefer to report vi for small  Truth-telling being dominant requires v’’i(s(v’’i , v-i)) + mi(v’’i , v-i) ≥ v’’i(s(vi , v-i)) + mi(vi , v-i) s(v’’i , v-i) = s(v’i , v-i) since setting x = s(v’i , v-i) maximizes v’’i(x) + ji vj(x) (This choice gives welfare , s(vi , v-i) gives 0, and other choices give - ) So, v’’i(s(v’i , v-i)) + mi(v’’i , v-i) ≥ v’’i(s(vi , v-i)) + mi(vi , v-i) From which we get by substitution: - ji vj(s(v’i , v-i)) +  + mi(v’’i , v-i) ≥ - ji vj(s(vi , v-i)) + mi(vi , v-i)  - ji vj(s(v’i , v-i)) +  + ji vj(s(v’’i , v-i)) + hi(v’’i, v-i) ≥ -ji vj(s(vi , v-i)) +ji vj(s(vi , v-i)) + hi(vi, v-i)   + hi(v’’i , v-i) ≥ hi(vi , v-i) Because s(v’’i , v-i) = s(v’i , v-i), by the logic of case 1, hi(v’’i , v-i) = hi(v’i , v-i) This gives  + hi(v’i , v-i) ≥ hi(vi , v-i) But by hypothesis we have hi(vi , v-i) > hi(v’i , v-i), so there is a contradiction for small  QED Other mechanisms might work too if v has special structure

Clarke tax “pivotal” mechanism Special case of Groves mechanism: hi(v-i) = - ji vj(s(v-i)) So, agent’s payment mi = ji vj(s(v)) - ji vj(s(v-i))  0 is a tax Intuition: Agent internalizes the negative externality he imposes on others by affecting the outcome Agent pays nothing if he does not change (“pivot”) the outcome Example: k=1, x1=”joint pool built” or “not”, mi = $ E.g. equal sharing of construction cost: -c / |A|, so vi(x1) = wi(x1) - c / |A| So, ui = vi (x1) + mi No pool Pool $0 ui =5 =10 u i =10 General preferences Quasilinear preferences

Clarke tax mechanism… Pros Cons Social welfare maximizing outcome Truth-telling is a dominant strategy Feasible in that it does not need a benefactor (i mi  0) Cons Budget balance not maintained (in pool example, generally i mi < 0) Have to burn the excess money that is collected Thrm. [Green & Laffont 1979]. Let the agents have quasilinear preferences ui(x, m) = mi + vi(x) where vi(x) are arbitrary functions. No social choice function that is (ex post) welfare maximizing (taking into account money burning as a loss) is implementable in dominant strategies If there is some party that has no private information to reveal and no preferences over x, welfare maximization and budget balance can be obtained by having that party’s payment be m0 = - i=1.. mi E.g. auctioneer could be agent 0 Might still not work if participation is voluntary Vulnerable to collusion Even by coalitions of just 2 agents

Implementation in Bayes-Nash equilibrium

Implementation in Bayes-Nash equilibrium Goal is to design the rules of the game (aka mechanism) so that in Bayes-Nash equilibrium (s1, …, s|A|), the outcome of the game is f(u1, …, u|A|) Weaker requirement than dominant strategy implementation An agent’s best response strategy may depend on others’ strategies Agents may benefit from counterspeculating each others’ preferences rationality endowments capabilities … Can accomplish more than under dominant strategy implementation E.g., budget balance & Pareto efficiency (social welfare maximization) under quasilinear preferences …

Expected externality mechanism [d’Aspremont & Gerard-Varet 79; Arrow 79] Like Groves mechanism, but sidepayment is computed based on agent’s revelation vi , averaging over possible true types of the others v-i Outcome (x1, x2, ..., xk, m1, m2, ..., m|A| ) Quasilinear preferences: ui(x, m) = mi + vi(x1, x2, ..., xk) Utilitarian setting: Social welfare maximizing choice Outcome s(v1, v2, ..., v|A| ) = maxx i vi(x1, x2, ..., xk) Others’ expected welfare when agent i announces vi is (vi) = v-i p(v-i) ji vj(s(vi , v-i)) Measures change in expected externality as agent i changes her revelation Thrm. Assume quasilinear preferences and statistically independent valuation functions vi. A utilitarian social choice function f: v -> (s(v), m(v)) can be implemented in Bayes-Nash equilibrium if mi(vi)= (vi) + hi(v-i) for arbitrary function h Unlike in dominant strategy implementation, budget balance achievable Intuitively, have each agent contribute an equal share of others’ payments Formally, set hi(v-i) = - [1 / (|A|-1)] ji (vj) Does not satisfy participation constraints (aka individual rationality) in general Agent might get higher expected utility by not participating

Myerson-Satterthwaite impossibility Avrim is selling a car to Tuomas, both risk neutral, quasilinear Each party knows his own valuation, but not the other’s The probability distributions are common knowledge Want a mechanism that is Ex post budget balanced Ex post Pareto efficient: Car changes hands iff vbuyer > vseller (Interim) individually rational: Both Avrim and Tuomas get higher expected utility by participating than not Thrm. Such a mechanism does not exist (even if randomized mechanisms are allowed) This impossibility is at the heart of more general exchange settings (NYSE, NASDAQ, combinatorial exchanges, …) !

Proof Seller’s valuation is sL w.p. a and sH w.p. (1-a) Buyer’s valuation is bL w.p. b and bH w.p. (1-b). Say bH > sH > bL > sL By revelation principle, can focus on truthful direct revelation mechanisms p(b,s) = probability that car changes hands given revelations b and s Ex post efficiency requires: p(b,s) = 0 if (b = bL and s = sH), otherwise p(b,s) = 1 Thus, E[p|b=bH] = 1 and E[p|b = bL] = a E[p|s = sH] = 1-b and E[p|s = sL] = 1 m(b,s) = expected price buyer pays to seller given revelations b and s Since parties are risk neutral, equivalently m(b,s) = actual price buyer pays to seller Since buyer pays what seller gets paid, this maintains budget balance ex post E[m|b] = (1-a) m(b, sH) + a m(b, sL) E[m|s] = (1-b) m(bH, s) + b m(bL, s)

Proof Individual rationality (IR) requires b E[p|b] – E[m|b]  0 for b = bL, bH E[m|s] – s E[p|s]  0 for s = sL, sH Bayes-Nash incentive compatibility (IC) requires b E[p|b] – E[m|b]  b E[p|b’] – E[m|b’] for all b, b’ E[m|s] – s E[m|s]  E[m|s’] – s E[m|s’] for all s, s’ Suppose a=b= ½, sL=0, sH=y, bL=x, bH=x+y, where 0 < 3x < y. Now, IR(bL): ½ x – [ ½ m(bL,sH) + ½ m(bL,sL)]  0 IR(sH): [½ m(bH,sH) + ½ m(bL,sH)] - ½ y  0 Summing gives m(bH,sH) - m(bL,sL)  y-x Also, IC(sL): [½ m(bH,sL) + ½ m(bL,sL)]  [½ m(bH,sH) + ½ m(bL,sH)] I.e., m(bH,sL) - m(bL,sH)  m(bH,sH) - m(bL,sL) IC(bH): (x+y) - [½ m(bH,sH) + ½ m(bH,sL)]  ½ (x+y) - [½ m(bL,sH) + ½ m(bL,sL)] I.e., x+y  m(bH,sH) - m(bL,sL) + m(bH,sL) - m(bL,sH) So, x+y  2 [m(bH,sH) - m(bL,sL)]  2(y-x). So, 3x  y, contradiction. QED

Social choice theory = preference aggregation = truthful voting

Social choice Collectively choosing among outcomes E.g. presidents Outcome can also be a vector E.g. allocation of money, goods, tasks, and resources Agents have preferences over outcomes Center knows each agent’s preferences Or agents reveal them truthfully by assumption Social choice function aggregates those preferences & picks outcome Outcome is enforced on all agents CS applications: Multiagent planning [Ephrati&Rosenschein], computerized elections [Cranor&Cytron], accepting a joint project, rating Web articles [Avery,Resnick&Zeckhauser], rating CDs...

Agenda paradox x y z x y z x y z Binary protocol (majority rule) = cup Three types of agents: x > z > y (35%) y > x > z (33%) z > y > x (32%) x y z x y z x y z Power of agenda setter (e.g. chairman) Vulnerable to irrelevant alternatives (z) Plurality protocol For each agent, most preferred outcome gets 1 vote Would result in x

Pareto dominated winner paradox Agents: x > y > b > a a > x > y > b b > a > x > y

Inverted-order paradox Borda rule with 4 alternatives Each agent gives 4 points to best option, 3 to second best... Agents: x=22, a=17, b=16, c=15 Remove x: c=15, b=14, a=13 x > c > b > a a > x > c > b b > a > x > c

Borda rule also vulnerable to irrelevant alternatives Three types of agents: Borda winner is x Remove z: Borda winner is y x > z > y (35%) y > x > z (33%) z > y > x (32%)

Majority-winner paradox Agents: Majority rule with any binary protocol: a Borda protocol: b=16, a=15, c=11 a > b > c b > c > a b > a > c c > a > b

Is there a desirable way to aggregate agents’ preferences? Set of outcomes O Each agent i has a most-to-least-preferred ordering Ri of O R = (R1, R2, ... , R|A| ) Social choice functional G (R, O ) = R To avoid unilluminating technicalities in proof, assume Ri and R are strict total orders Some possible (weak) desiderata of G 1. Pareto principle: If every agent prefers x to y , then x is preferred to y in R 2. Independence of irrelevant alternatives: If x is preferred to y in G (R, O ), and if R’ is another preference profile s.t. each agent’s preference between x and y is the same as in R, then x is preferred to y in G (R’, O ) 3. Nondictatorship: No agent is decisive for every pair of outcomes in O Arrow’s impossibility theorem: If |O | ≥ 3, then no G satisfies desiderata 1-3

Proof of Arrow’s theorem (1 of 3) Follows [Mas-Colell, Whinston & Green, 1995] Assuming G is Paretian and independent of irrelevant alternatives, we show that G is dictatorial Def. Set S  A is decisive for x over y whenever x >i y for all i  S x < i y for all i  A-S => x > y Lemma 1. If S is decisive for x over y, then for any other candidate z, S is decisive for x over z and for z over y Proof. Let S be decisive for x over y. Consider: x >i y >i z for all i  S and y >i z >i x for all i  A-S Since S is decisive for x over y, we have x > y Because y >i z for every agent, by the Pareto principle we have y > z Then, by transitivity, x > z By independence of irrelevant alternatives (y), x > z whenever every agent in S prefers x to z and every agent not in S prefers z to x. I.e., S is decisive for x over z To show that S is decisive for z over y, consider: z >i x >i y for all i  S and y >i z >i x for all i  A-S Then x > y since S is decisive for x over y z > x from the Pareto principle and z > y from transitivity Thus S is decisive for z over y 

Proof of Arrow’s theorem (2 of 3) Given that S is decisive for x over y, we deduced that S is decisive for x over z and z over y. Now reapply Lemma 1 with decision z over y as the hypothesis and conclude that S is decisive for z over x which implies (by Lemma 1) that S is decisive for y over x which implies (by Lemma 1) that S is decisive for y over z Thus: Lemma 2. If S is decisive for x over y, then for any candidates u and v, S is decisive for u over v (i.e., S is decisive) Lemma 3. For every S  A, either S or A-S is decisive (not both) Proof. Suppose x >i y for all i  S and y >i x for all i  A-S (only such cases need to be addressed, because otherwise the left side of the implication in the definition of decisiveness between candidates does not hold). Because either x > y or y > x, S is decisive or A-S is decisive 

Proof of Arrow’s theorem (3 of 3) Lemma 4. If S is decisive and T is decisive, then S  T is decisive Proof. Let S = {i: z >i y >i x }  {i: x >i z >i y } Let T = {i: y >i x >i z }  {i: x >i z >i y } For i  S  T, let y >i z >i x Now, since S is decisive, z > y Since T is decisive, x > z Then by transitivity, x > y So, by independence of irrelevant alternatives (z), S  T is decisive for x over y. (Note that if x >i y, then i  S  T.) Thus, by Lemma 2, S  T is decisive  Lemma 5. If S = S1  S2 (where S1 and S2 are disjoint and exhaustive) is decisive, then S1 is decisive or S2 is decisive Proof. Suppose neither S1 nor S2 is decisive. Then ~ S1 and ~ S2 are decisive. By Lemma 4, ~ S1  ~ S2 = ~S is decisive. But we assumed S is decisive. Contradiction  Proof of Arrow’s theorem Clearly the set of all agents is decisive. By Lemma 5 we can keep splitting a decisive set into two subsets, at least one of which is decisive. Keep splitting the decisive set(s) further until only one agent remains in any decisive set. That agent is a dictator. QED

Stronger version Arrow’s theorem In Arrow’s theorem, social choice functional G outputs a ranking of the outcomes The impossibility holds even if only the highest ranked outcome is sought: Thrm. Let |O | ≥ 3. If a social choice function f: R -> outcomes is monotonic and Paretian, then f is dictatorial f is monotonic if [ x = f(R) and x maintains its position in R’ ] => f(R’) = x x maintains its position whenever x >i y => x > i’ y Proof. From f we construct a social choice functional F that satisfies the conditions of Arrow’s theorem For each pair x,y of outcomes in turn, to determine whether x > y in F, move x and y to the top of each voter’s preference order don’t change their relative order order of other alternatives can change Lemma 1. If R’ and R’’ are constructed from R by moving a set X of outcomes to the top in this way, then f(R’) = f(R’’) Proof. Because f is Paretian, f(R’)  X. Thus f(R’) maintains its position in going from R’ to R’’. Then, by monotonicity of f, we have f(R’) = f(R’’) Note: Because f is Paretian, we have f = x or f = y (and by lemma 1 not both) F is transitive (total order) (we omit proving this part) F is Paretian (if everyone prefers x over y, then x gets chosen and vice versa) F satisfies independence of irrelevant alternatives (immediate from lemma 1) By earlier version of the impossibility, F (and thus f) must be dictatorial. QED

Mechanism design (strategic voting) Tuomas Sandholm Associate Professor Computer Science Department Carnegie Mellon University

Goal of mechanism design Implementing a social choice function f(R) using a game Actually, say we want to implement f(u1, …, u|A|) Center = “auctioneer” does not know the agents’ preferences Agents may lie unlike in the theory of social choice which we discussed in class before Goal is to design the rules of the game (aka mechanism) so that in equilibrium (s1, …, s|A|), the outcome of the game is f(u1, …, u|A|) Mechanism designer specifies the strategy sets Si and how outcome is determined as a function of (s1, …, s|A|)  (S1, …, S|A|) Variants Strongest: There exists exactly one equilibrium. Its outcome is f(u1, …, u|A|) Medium: In every equilibrium the outcome is f(u1, …, u|A|) Weakest: In at least one equilibrium the outcome is f(u1, …, u|A|)

Revelation principle Any outcome that can be supported in Nash (dominant strategy) equilibrium via a complex “indirect” mechanism can be supported in Nash (dominant strategy) equilibrium via a “direct” mechanism where agents reveal their types truthfully in a single step Agent 1’ s preferences Agent |A|’ . Strategy formulator Original “complex” “indirect” mechanism Outcome Constructed “direct revelation” mechanism

Uses of the revelation principle Literal: “Only direct mechanisms needed” Problems: Strategy formulator might be complex Complex to determine and/or execute best-response strategy Computational burden is pushed on the center (assumed away) Thus the revelation principle might not hold in practice if these computational problems are hard This problem traditionally ignored in game theory Even if the indirect mechanism has a unique equilibrium, the direct mechanism can have additional bad equilibria As an analysis tool Best direct mechanism gives tight upper bound on how well any indirect mechanism can do Space of direct mechanisms is smaller than that of indirect ones One can analyze all direct mechanisms & pick best one Thus one can know when one has designed an optimal indirect mechanism (when it is as good as the best direct one)