VOTING GAMES Prof. Yeşim Kuştepeli ECO 4413 Game Theory 1

O UTLINE Choosing Among Voting Mechanisms Majority Rule with Two Choices Plurality Rule and the Condorcet Candidate A Modified Plurality Rule: The Single Transferable Vote Strategic Voting with Plurality Rule 2 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

The individual is a consumer: individuals demand a variety of consumption goods and services from which they drive welfare. The individual provides productive services: the most obvious resource provided by the individual is labor. The individual participates in the political process: by voting and other political activities, the individual expresses his or her preferences regarding the government’s provision of goods and services. 3 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

The acceptance or rejection by a corporation’s broad of directors of merger offer tendered by competing firm. The acceptance or rejection by a union’s rank and file of the labor contract negotiated by union’s officers. The acceptance or rejection by a state’s voters of proposal to issue new bonds with which to build new roads and bridges. 4 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

The outcome of any vote depends on the voting rules and the ballots submitted by voters. Voting is a type of game and amenable to game theoretical analysis. If the secret ballots are used, then the voters are playing a static game. Strategically sophisticated voter realize that it may not always be optimal to vote sincerely. 5 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

T WO DIFFERENT MODELS OF VOTER : Naive voter: he always casts his ballot honestly, even when it is not in his best interest to do so. Strategic voter: she always casts her ballot so as to obtain the best outcome possible given her information, the voting rules, and her beliefs about how the other voters will behave. 6 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

C HOOSING A MONG V OTING M ECHANISMS Plurality voting: ballots are prepared which voters choose only one of the candidate and vote for him. After the election, these ballots are tallied (counted) and the person receiving the largest number of votes is declared the winner. This voting mechanism most commonly used to make public decision or to elect political candidates. 7 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

Approval voting: voters could be allowed to vote for as many candidates as they approve of. The ballots are tallied and the candidate receiving the highest number of approval votes is declared the winner. Social choice mechanism (SCM): voting is just one procedure the members of a group could use to from a “collective” ranking of a group of candidates on the basis of their individual rankings. We will refer to any such mechanism as a Social choice mechanism 8 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

A RROW ’ S IMPOSSIBILITY THEOREM : There is no social choice mechanism that simultaneously satisfies the Pareto condition, the independence of irrelevant alternatives condition, and the transitivity with unlimited domain condition, and is also non-dictatorial. 9 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

A RROW ’ S REQUIREMENTS : 1) No candidate is ranked higher by every member of the group than the candidate ranked highest by SCM. 2) The ranking provided by the SCM is always complete and transitive whenever the ranking of every member of the group is complete and transitive. 3) The relative ranking of any two candidates provided by the SCM depends on only the individual rankings. It does not depend on how these two candidates are ranked against any other candidates. 4) The SCM is not a dictatorship. There is no individual whose ranking always matches the ranking of the SCM, whatever the ranking of the other members of the group 10 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

M AJORITY R ULE WITH T WO C HOICES Majority rule: the winner is the candidate who receives more than half of the votes cast. And the winner of election is that candidate who receives the most votes. In an election between two candidates decided by a majority rule, all voters will rationally abstain or vote honestly for their most preferred candidate. 11 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

W E WILL STUDY THE BEHAVIOR OF VOTING RULES ASSUMING THAT : all voters are well informed their preferences are common knowledge they take voting mechanism the slate of candidates as given 12 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

C ANDIDATE AND VOTER LOCATIONS ON LEFT - TO - RIGHT SCALE 13 Prof. Yeşim Kuştepeli ECO 4413 Game Theory 0,39 0,35 0,3010 Location of candidate A Location of candidate B Location of voter

Candidate Position Game: the candidates are assumed to choose their positions simultaneously and cannot budge from that choice once it is made. We will also suppose that voter preferences and candidates objectives are common knowledge. We will refer to the resulting game as the Candidate Position Game. Positioning at 0.50 is a weakly dominant strategy for both candidates, and so both candidates positioning themselves at This is also the unique Nash equilibrium for this game is the position of the median voter. candidates’ positions converge to the position of the median voter. 14 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

The Median Voter Theorem: if there are two candidates, the two candidates’ political positions can be represented by their location on a continuous linear scale, each voter’s preference over these positions is singe-peaked, the distribution of voter preferences is common knowledge, and the election is decided by majority rule; then the unique Nash equilibrium strategy in the Candidate position game is both candidates to position themselves at the median voter’s position. 15 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

D ISTRIBUTION OF V OTER PREFERENCES 16 Prof. Yeşim Kuştepeli ECO 4413 Game Theory Voter position Number of voters

P LURALITY R ULE AND THE C ONDORCET C ANDIDATE A candidate who is preferred by a majority of voters to any other alternative in a series of pairwise comparisons is called condorcet candidate after the Marquis de Condorcet (French mathematician and philosopher ). 17 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

H YPOTHETICAL V OTER P REFERENCE 18 Prof. Yeşim Kuştepeli ECO 4413 Game Theory Voter type Number of voters with that type preference ABCABC b > d > e d > b > e e > b > d Cyclical majority: no one candidate can muster a majority of votes against all alternatives, hence there is no Condorcet candidate.

Voter Preferences That Produce No Condorcet Candidate 19 Prof. Yeşim Kuştepeli ECO 4413 Game Theory Voter type Number of voters with that type preference ABCABC x > y > z y > z > x z > x > y

Single transferable vote: voters are given a slate of candidates and are asked to vote for one. if no one candidate receives a majority of the votes cast, a second ballot is taken in which the candidate receiving the smallest number of votes from the first ballot dropped from the list of eligible candidates. 20 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

S HAREHOLDER P REFERENCES 21 Prof. Yeşim Kuştepeli ECO 4413 Game Theory Voter Number of Shares/voters Preference ABCDABCD z > y > x x > z > y y > x > z y > z > x

T HE O UTCOME OF THE E LECTION A S B’ S V OTE C HANGES W HEN THE O THER V OTERS A RE H ONEST 22 Prof. Yeşim Kuştepeli ECO 4413 Game Theory Voter B’s BallotWinner x>z>y (honest) x>y>z z>x>y z>y>x y>x>z y>z>x ZYZZYYZYZZYY

S TRATEGIC V OTING WITH P LURALITY R ULE Chairman’s paradox: one member of a committee -the chairman – is given the power to break ties. the resulting plurality-rule voting game has a unique iterated dominant strategy equilibrium whose outcome is the one considered the worst by the “powerful” chairman. 23 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

AN EXAMPLE Three professors at economics department, Ms. Ayşe, Ms. Fatma, and Mr. Ali, have been asked by the economics department to choose the mathematics requirement for economics major. There are considering three options.  require no math courses for students majoring in economics. L (low requirement)  require one term of Univariate calculus. M (medium requirement)  require two terms of calculus: one of Univariate calculus and one of multivariate. H (high requirement) 24 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

V OTER P REFERENCES 25 Prof. Yeşim Kuştepeli ECO 4413 Game Theory voterpreferences Ms. Ayşe Mr. Ali Ms. Fatma L > M >H H > L >M M > H >L One of the committee members, Ms. Ayşe, has been designated the Chair of committee. The chair carries the power to cast a tie- breaking vote. It is more likely that Ms. Ayşe’s preferred outcome will be chosen. This institution is confirmed by the naïve model.

T HE N AÏVE V OTING M ODEL 26 Prof. Yeşim Kuştepeli ECO 4413 Game Theory If all three members of the committee vote naively, then the outcome of vote is straightforward and intuitive: Ms. Ayşe  L Ms. Fatma  M Mr. Ali  H L is chosen by the committee and economics majors will not be required to take any math course.

T HE S TRATEGIC V OTING M ODEL Strategic voting behavior: the game essentially reduces to a static game in which the set of strategies and the set of moves coincide. The concept we will employ to make predictions about Strategic voting behavior is Nash equilibrium. But, if we can find dominant strategies and iterated dominant strategies, we will use them as our predictors. Because of Ms. Ayşe’s voting power, it is likely that he has dominated strategies that can be removed from the game. 27 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

Consider Ms. Ayşe’s best response to every possible pair of votes by Ms. Fatma and Mr. Ali. If Ms. Fatma and Mr. Ali vote for the same candidate, then Ms. Ayşe s vote is irrelevant and all three of Ms. Ayşe’s strategies are equally good. If Ms. Fatma and Mr. Ali split their vote, then Ms. Ayşe’s vote determines the outcome. Hence Ms. Ayşe has a weakly dominant strategy of voting for L. we would predict, and so presumably would Ms. Fatma and Mr. Ali, that he will vote for L. 28 Prof. Yeşim Kuştepeli ECO 4413 Game Theory

P AYOFF M ATRIX ASSUMING M S. A YŞE VOTES FOR L 29 Prof. Yeşim Kuştepeli ECO 4413 Game Theory Ms. Fatma’s vote Mr. Ali’s vote HML H(Best, Middle, worst)(Middle, worst best,)(Middle, worst, best) M(middle, worst, best)(worst, Best, middle)(Middle, worst, best) L(middle, Worst, best)(Middle, Worst, best) Payoffs: Mr. Ali, Ms. Fatma and Ms. Ayşe

P AYOFF M ATRIX ASSUMING M S. A YŞE VOTES FOR L AND M S. F ATMA DOES NOT VOTE FOR L 30 Prof. Yeşim Kuştepeli ECO 4413 Game Theory Ms. Fatma’s vote Mr. Ali’s vote HM H(Best, Middle, worst)(Middle, worst best,) M(middle, worst, best)(worst, Best, middle) L(middle, Worst, best)(Middle, Worst, best) Payoffs: Mr. Ali, Ms. Fatma and Ms. Ayşe

P AYOFF M ATRIX ASSUMING M S. A YŞE VOTES FOR L, M S. F ATMA DOES NOT VOTE FOR L, AND M R. A LI VOTES FOR H 31 Prof. Yeşim Kuştepeli ECO 4413 Game Theory Ms. Fatma’s vote Mr. Ali’s vote HM H (Best, Middle, worst) (Middle, worst best,) Payoffs: Mr. Ali, Ms. Fatma and Ms. Ayşe

H wins, even though the Chair of the committee, Ms. Ayşe, considers this his worst outcome. The “power” to break ties turns out to be a bad thing for the person who is unlucky enough to be saddled with it. Oddly enough, once the committee is assigned, there should be a fight among the members not to be chosen as Chair, even though the Chair brings with it extra voting “privileges.” Of course, what drives the result is that Ms. Fatma does not vote “naively”. she recognizes that if she were to do so, then she would end up with what she considers to be the worst outcome. Therefore, she votes strategically. 32 Prof. Yeşim Kuştepeli ECO 4413 Game Theory