Empir iric ical l Aspects of Plu Pluralit lity Ele lectio ions - PowerPoint PPT Presentation

Empir iric ical l Aspects of Plu Pluralit lity Ele lectio ions David R. M. Thompson, Omer Lev, Kevin Leyton-Brown & Jeffrey S. Rosenschein COMSOC 2012 Kraków, Poland

What is is a ( pure) Nash Equilib ilibriu ium? m? (pur A solution concept involving games where all players know the strategies of all others. If there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium. Adapted from Roger McCain’s Game Theory: A Nontechnical Introduction to the Analysis of Strategy

What is is a Nash Equilib ilibriu ium? m? Example le: votin ing pris isoners rs’ dile ilemma mma… Everett Pete Delmar 1 st Escape Riot Stay in preference prison 2 nd Riot Escape Escape preference 3 rd Stay in Stay in Riot preference prison prison

What is is a Nash Equilib ilibriu ium? m? Example le: votin ing pris isoners rs’ dile ilemma mma… Everett Pete Delmar Suppose tie is broken by 1 st Escape Riot Stay in preference deciding to stay in prison prison 2 nd Riot Escape Escape preference 3 rd Stay in Stay in Riot preference prison prison

What is is a Nash Equilib ilibriu ium? m? Example le: votin ing pris isoners rs’ dile ilemma mma… Everett Pete Delmar 1 st Escape Riot Stay in preference prison 2 nd Riot Escape Escape preference 3 rd Stay in Stay in Riot preference prison prison

What is is a Nash Equilib ilibriu ium? m? Example le: votin ing pris isoners rs’ dile ilemma mma… Everett Pete Delmar 1 st Escape Riot Stay in preference prison 2 nd Riot Escape Escape preference 3 rd Stay in Stay in Riot preference prison prison

What is is a Nash Equilib ilibriu ium? m? Example le: votin ing pris isoners rs’ dile ilemma mma… Everett Pete Delmar 1 st Escape Riot Stay in preference prison 2 nd Riot Escape Escape preference 3 rd Stay in Stay in Riot preference prison prison

What is is a Nash Equilib ilibriu ium? m? Example le: votin ing pris isoners rs’ dile ilemma mma… Everett Pete Delmar But if players are not truthful, weird things can 1 st Escape Riot Stay in preference prison happen… 2 nd Riot Escape Escape preference 3 rd Stay in Stay in Riot preference prison prison

What is is a Nash Equilib ilibriu ium? m? Example le: votin ing pris isoners rs’ dile ilemma mma… Everett Pete Delmar 1 st Escape Riot Stay in preference prison 2 nd Riot Escape Escape preference 3 rd Stay in Stay in Riot preference prison prison

Problem 1: � Can we decrease the number of pure Nash equilibria? � (especially eliminating the senseless ones…) �

The truthfuln lness in incentiv ive Each player’s utility is not just dependent on the end result, but players also receive a small 𝜁 when voting truthfully . The incentive is not large enough as to influence a voter’s choice when it can affect the result.

The truthfuln lness in incentiv ive Example le Everett Pete Delmar 1 st Escape Riot Stay in preference prison 2 nd Riot Escape Escape preference 3 rd Stay in Stay in Riot preference prison prison

Problem 2: � How can we identify pure Nash equilibria? �

Actio ion Graph Game mes A compact way to represent games with 2 properties: Anonymity: Context specific A>B A B payoff independence: depends on payoff depends own action on easily A B and number of calculable statistic players for summing other each action. actions. A B B>A Calculating the equilibria using Support Enumeration Method (worst case exponential, but thanks to heuristics, not common).

� Now we have a way to find pure equilibria, and a way to ignore absurd ones. � So? �

The scenario io 5 candidates & 10 voters. Voters have Borda-like utility functions (gets 4 if favorite elected, 3 if 2 nd best elected, etc.) with added truthfulness incentive of 𝜁 =10 -6 . They are randomly assigned a preference order over the candidates. This was repeated 1,000 times.

Result lts: numb mber r of equilib ilibria ia 1 ¡ 0.9 ¡ 0.8 ¡ All ¡games ¡ 0.7 ¡ Share ¡of ¡experiments ¡ 0.6 ¡ Games ¡with ¡ 0.5 ¡ true ¡as ¡NE ¡ 0.4 ¡ Games ¡without ¡ 0.3 ¡ true ¡as ¡NE ¡ 0.2 ¡ 0.1 ¡ 0 ¡ 0 ¡ 5 ¡ 10 ¡ 15 ¡ 20 ¡ 25 ¡ 30 ¡ 35 ¡ 40 ¡ 45 ¡ 50 ¡ 55 ¡ 60 ¡ 65 ¡ 70 ¡ 75 ¡ 80 ¡ 85 ¡ 90 ¡ 95 ¡100 ¡ 105 ¡ 110 ¡ 115 ¡ 120 ¡ 125 ¡ 130 ¡ 135 ¡ 140 ¡ 145 ¡ Number ¡of ¡PSNE ¡ In 63.3% of games, voting truthfully was a Nash equilibrium. 96.2% have less than 10 pure equilibria (without permutations). 1.1% of games have no pure Nash equilibrium at all.

Result lts: type of equilib ilibria ia truthful 600 ¡ 500 ¡ Condorcet ¡ NE ¡ 400 ¡ Number ¡of ¡equlibria ¡ Truthful ¡NE ¡ 300 ¡ Non ¡ 200 ¡ truthful/ Condorcet ¡ NE ¡ 100 ¡ 0 ¡ 0 ¡ 1 ¡ 2 ¡ 3 ¡ 4 ¡ 5 ¡ 6 ¡ 7 ¡ 8 ¡ 9 ¡ 10 ¡ 11 ¡ 12 ¡ 13 ¡ 14 ¡ 15 ¡ 16 ¡ 17 ¡ 18 ¡ 19 ¡ 20 ¡ 21 ¡ 22 ¡ 23 ¡ 24 ¡ 25 ¡ Number ¡of ¡PSNE ¡for ¡each ¡experiments ¡ 80.4% of games had at least one truthful equilibrium. Average share of truthful-outcome equilibria: 41.56% (without incentive – 21.77%).

Result lts: type of equilib ilibria ia Con Condor dorce cet 600 ¡ 500 ¡ Condorcet ¡ NE ¡ 400 ¡ Number ¡of ¡equlibria ¡ Truthful ¡NE ¡ 300 ¡ Non ¡ 200 ¡ truthful/ Condorcet ¡ NE ¡ 100 ¡ 0 ¡ 0 ¡ 1 ¡ 2 ¡ 3 ¡ 4 ¡ 5 ¡ 6 ¡ 7 ¡ 8 ¡ 9 ¡ 10 ¡ 11 ¡ 12 ¡ 13 ¡ 14 ¡ 15 ¡ 16 ¡ 17 ¡ 18 ¡ 19 ¡ 20 ¡ 21 ¡ 22 ¡ 23 ¡ 24 ¡ 25 ¡ Number ¡of ¡PSNE ¡for ¡each ¡experiments ¡ 92.3% of games had at least one Condorcet equilibrium. Average share of Condorcet equilibrium: 40.14%.

Result lts: socia ial l welf lfare average rank 0.6 ¡ All ¡Games ¡(with ¡ 0.5 ¡ truthfulness-­‑ incenJve) ¡ Ignoring ¡ Average ¡percentage ¡of ¡equilibra ¡ Condorcet ¡ 0.4 ¡ winners ¡ Ignoring ¡truthful ¡ winners ¡ 0.3 ¡ Without ¡ truthfulness ¡ 0.2 ¡ incenJve ¡ 0.1 ¡ 0 ¡ [0,1) ¡ [1,2) ¡ [2,3) ¡ [3,4) ¡ 4 ¡ Average ¡ranking ¡(upper ¡value) ¡ 71.65% of winners were, on average, above median. 52.3% of games had all equilibria above median.

lfare raw sum Result lts: socia ial l welf 92.8% of games, there was no pure equilibrium with the worst result (only in 29.7% was best result not an equilibrium). 59% of games had truthful voting as best result (obviously dominated by best equilibrium).

But what about more common situations, when we don’t have full information? �

Bayes-Nash equilib ilibriu ium Each player doesn’t know what others prefer, but knows the distribution according to which they are chosen . So, for example, Everett and Pete don’t know what Delmar prefers, but they know that: 50% 45% 5% 0% Stay in Stay in 1 st Escape Riot prison prison preference Stay in 2 nd Riot Escape Escape prison preference 3 rd Stay in preference Riot Riot Escape prison

Bayes-Nash equilib ilibriu ium scenario io 5 candidates & 10 voters. We choose a distribution: assign a probability to each preference order. To ease calculations – only 6 orders have non-zero probability. We compute equilibria assuming voters are chosen i.i.d from this distribution. All with Borda-like utility functions & truthfulness incentive of 𝜁 =10 -6 . This was repeated 50 times.

Result lts: numb mber r of equilib ilibria ia 40 35 # of equilibria (with 30 truthfulness) 25 20 15 10 5 0 0 10 20 30 40 # of equilibria (without truthfulness) Change (from incentive-less scenario) is less profound than in the Nash equilibrium case (76% had only 5 new equilibria).

Result lts: type of equilib ilibria ia truthful not truthful 10.6 9.52 5 4.84 4.84 0.7 0.7 0.02 0.02 0 1 candidate 2 candidates 3 candidates 4 candidates 5 candidates 95.2% of equilibria had only 2 or 3 candidates involved in the equilibria. Leading to…

Result lts: proposit itio ion In a plurality election with a truthfulness incentive of 𝜁 , as long as 𝜁 is small enough, for every c 1 , c 2 ∈ C either c 1 Pareto dominates c 2 (i.e., all voters rank c 1 higher than c 2 ), or there exists a pure Bayes-Nash equilibrium in which each voter votes for his most preferred among these two candidates.

Pr Proof sketch c 1 Suppose I prefer c 1 to c 2 . … c 2 If it isn’t Pareto-dominated, there is a probability c 2 P that a voter would prefer c 2 over c 1, and … c 1 hence P n/2 that my vote would be pivotal. If 𝜁 is small enough, so one wouldn’t be tempted to vote truthfully, each voter voting for preferred type of c 1 or c 2 is an equilibrium