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Two Player Game Theory
The current voting system in the United States, the plurality vote - a single vote ballot where the
candidate who receives the most votes wins, - contains many problems. For example, it is easily
manipulated and does not always select the winner most accurate of the views of the general public.
Even the claimed improved methods, Instant Runoff Voting and Single Transferable Voting, that have
been implemented in in San Francisco, Portland, ME1, and Australia2 still possess many flaws and
inaccuracies. Both of these two “improved” methods lack the voting method characteristic of Voter
Fairness. Voter Fairness, also known as Monotonicity, guarantees that when a voter votes in favor of a
selected candidate, it will help that candidate and when a voter votes against a candidate, it will hurt
that candidate. In the methods currently implemented in San Francisco and a few other cities, voting for
a candidate who the voter wants to win actually could potentially hurt that candidate.3 The lack of
Voter Fairness in these methods is one major flaw with these innovative methods. Two Player Game
Theory is a vote counting method that would eliminate many of these flaws that are prevalent in the
current voting system and mitigate one’s ability to manipulate an election.
To understand the method of Two Player Game Theory, some basic game theory concepts need
to be grasped. In the two player game of “Rock, Paper, Scissors”, the players choose either “Rock”,
“Paper”, or “Scissors”. If rock and paper are played, then the player who played paper wins, because
paper covers rock. If paper and scissors are played, then the player who played scissors wins, because
scissors cuts paper. If rock and scissors are played, then the player who played rock wins, because rock
crushes scissors. If the same choices are played, then the game results in a tie. As shown in Figure 1, a
cycle is created among the winners. This cycle indicates that there is no choice that beats all of the other
1
Egelko, B. (2010, April 20). S.f. instant-runoff voting upheld. The Chronicle. Retrieved from http://articles.sfgate.com/2010-0420/bay-area/20856835_1_instant-runoff-voting-voting-system-instant-runoffs
2
Tideman, N. (1995). The single transferable vote. The Journal of Economic Perspectives, 9(1), 27-38.
3
Quas, A. (2004). Anomalous outcomes in preferential voting. Stochatics and Dynamics, 4(1), 95-105.
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choices. Every choice wins against one of the choices and loses to the other. This cycle provides the
entertaining level of the game, because no choice beats all.
Figure 1: Diagram of "Rock, Paper,
Scissors". The arrows point in the
direction of the winning choice.
If the rules of the game were changed to that rock beat paper too, then the game would lose its
entertaining aspect because both players, assuming that they understand that they could not lose by
playing rock, would play rock one hundred percent of the time. The game would end with a tie all of the
time. These two players are applying an aspect of game theory known as equilibrium strategy.
Equilibrium strategy is a game theory concept that in games like “rock, paper, scissors”, where there are
head to head match ups and a clear winner in each match up, that there is a certain percentage for the
amount of times one should play each move. This equilibrium strategy would, on average, win or tie
with every other strategy. In the example where rock beat all of the choices, the equilibrium strategy
would be to play rock all of the time and never play paper or scissors. This strategy would win whenever
the opposing player played any of the other choices and tie whenever the player played rock. The
equilibrium strategy in the original version of “Rock, Paper, Scissors” would be to play each choice one
third of the time because every choice has the same number of wins and losses. These are the only
possible two equilibrium strategies when three or fewer choices are available.
Now What if the number of choices in this game were increased? How would this change the
equilibrium strategy? If four choices are available, it turns out that there are only two equilibrium
strategies for this game. The first strategy is to use one of the choices all of the time and never use the
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other choices (Figure 2). The other strategy is to use three of choices each a third of the time (Figure 3).
These are the same two possible equilibrium strategies for when there are only three or fewer choices.
When there are five or six choices, two new equilibrium strategies are formed. With six or fewer choices
there are two more possible equilibrium strategies, one of the strategies is to play five of the choices
each one fifth of the time (Figure 4), and the other strategy is to play two of the choices one third of the
time and three of the choices each one ninth of the time (Figure 5). Any combination of head to head
match ups, when only six or fewer choices are provided, will give one of these four strategies as the
equilibrium strategy for the game.
Figure 2: If one choice beats all of the
other choices in head to head match
ups then the equilibrium strategy is to
play that choice all of the time.
Figure 3: A possible equilibrium strategy for
a game with head to head match ups is to
play three of the choices each one third of
the time.
Figure 4: Five choices where each choice
should be played each one fifth of the time is
another possible equilibrium strategy for a
game with five or six choices.
Figure 5: Another possible equilibrium strategy for a
game with six or fewer choices, the Bicycle equilibrium
strategy, occurs when there is a cycle within a cycle.
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Now instead of the choices being choices in a game, what if the choices were candidates for an
election? The United States’ current system of a single vote ballet with the most votes win or the
Plurality vote, consists of many flaws that allow for the system to be manipulated and does not always
elect the person best representative of the population. One problem with the Plurality Voting System is
that there is a chance that even though a candidate would beat all of the other candidates in head to
head match ups, this candidate would not win the election. A candidate, who is able to win all of the
head to head match ups with all of the other candidates, is known as a Condorcet winner. Having a
Condorcet winner actually occurs more often than not. A Condorcet winner occurs 90.57% of the time
when 3 candidates are in the election, 82.80% of the time with 4 candidates, 75.10% with 5 candidates,
and 68.01% with 6 candidates.4 However in the Plurality voting method, despite a candidate being a
Condorcet winner, it could still potentially could lose the election. The lack of electing a Condorcet
winner in an election is a major flaw in the Plurality voting method and in many other voting methods.
The first major change to improve the voting system and elect the candidate best representative
of the population is to switch from a single vote ballot to a preferential voting ballot. The preferential
ballot is a voting ballot that instead of just selecting the voter’s top choice for the election, the voter
ranks all of the candidates from first choice to last choice. The ranking system of the preferential ballot
provides more information about the voters and allows for more accurate voting methods to be used to
elect the winner that best represents the voting population opinions. The preferential ballot calculates
all of the head to head match ups. These calculations of head to head match ups allow for diagrams
similar to “Rock, paper, Scissors” (Figure 1) to be constructed. These diagrams display if a candidate
would win or lose when compared to another candidate. It also shows if there is a Condorcet winner
4
Tabachnik, M. (2011, April 25). Strategy and Effectiveness: An Analysis of Preferential Ballot Voting Methods.
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(Figure 6). If a candidate wins all of the head to head match ups with all of the other candidates, then
the candidate is a Condorcet winner.
Figure 6: A diagram of five candidates where Candidate A beats
Candidates B, C, D, and E in head to head match ups and therefore
Candidate A is a Condorcet winner.
In the rare occasions when there is not a Condorcet winner, there will be a resulting cycle in the
diagram. This cycle is known as Arrow’s paradox.5 Figure 7 illustrates one possible diagram where
Arrow’s Paradox occurs. According to Figure 7, if the voting population were given the choice between
Canidate A or Candidate B, the majority would choose A. When given the choice between Candidates B
and C, the voting population would choose B, but when given the choice between Candidates C and A,
the voting population would choose C. This creates a cycle in the diagram and does not allow for a
Condorcet winner. For these rare occasions, there needs to be another method to designate who wins
the election.
5
Bernholz, P. (1973). Logrolling, arrow paradox and cyclical majorities . Public Chocie, 15(1), 87-95.
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Figure 7: A diagram of four candidates where
there is not a Condorcet winner.
If the election were treated like a game and an equilibrium strategy was calculated for game,
then not only would method always select the Condorcet winner, if there was one, but it would also
provide for a method to figure out who to elect ,when there is not a Condorcet winner. If there was a
Condorcet winner the equilibrium solution for election would be that the Condorcet would receive a
one – or be recommended to play all of the time – and all of the other candidates would receive zeros
(Figure 2).
When there is not a Condorcet winner, then the equilibrium strategy will consist of more than
one candidate. There are two possible ways to determine the winner from the equilibrium strategy,
when there is not a Condorcet winner. One method is to select the winner by chance, with the
probability of that Candidate is selected is the corresponding probability in the equilibrium strategy. For
example, if the equilibrium strategy was that of Figure 4, then each candidate would have a one fifth
chance of being selected as the winner. The other method is to not elect one winner but rather a group
of winners. If the candidate received a proportion in the equilibrium strategy then that candidate would
be selected in the group of winners. The group of winners would then vote on all of the decisions with
the proportion of the equilibrium strategy that each candidate receives is the proportion of the vote the
candidate has. For example if the equilibrium strategy was that of Figure 4, then there would be five
candidates elected and each would have one fifth of the vote for making decisions. If the equilibrium
strategy was that of Figure 5, then five candidates would be in the group with two of the candidates
having one third of the vote and the other three each having one ninth of the vote. The equilibrium
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strategy also turns out to have it where not winner is ever irrelevant. This means that if there was a
committee formed from the equilibrium strategy and each one had the proportion of the vote that
corresponded to their proportion in the equilibrium strategy then there would never be a case where a
winner had over half of the votes.
This method, Two Player Game Theory, also has the property of Voter Fairness. Voter Fairness is
the property that voting for a candidate helps the candidate and voting against the candidate hurts the
candidate. In the preferential ballot, it translates to that whenever the voter increases the ranking of a
candidate it helps the candidate and whenever the voter decreases the ranking of a candidate it hurts
the candidate. It is a fundamental property that is not possessed by the claimed improved methods
implemented in San Francisco and Portland, ME.
Two Player Game Theory also possesses the property of Clone invariance. Clone invariance is
the characteristic that if a clone of a candidate, a candidate who had all of the same opinions as another
candidate, were to enter the election that it would not hurt the candidates. The Plurality vote is not
clone invariant because as seen in the past, if a similar candidate were to enter the race then it would
hurt the other candidate. This has been experienced in the 1992 election when Ross Perrot stole
potential votes away from Bush, in the 2000 election when Nader took potential votes from Gore, and in
the 1912 election when Theodore Roosevelt took votes away from Taft and this allowed Woodrow
Wilson to win the election. Two Player Game Theory is clone invariant, because if the in the preferential
the candidates would still be placed next to each other and would not affect when they were compared
in head to head match up with the other candidates. This property of clone invariance, helps mitigate
one’s ability to manipulate an election.
The Two Player Game Theory method also has the property of loser independence. Loser
independence is the property that no party outside of the people in the cycle could affect the outcome
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of the election. In other words if a losing candidate were to drop out of the election it would not affect
the outcome.6 In Two Play Game Theory, if a losing candidate or a candidate that was not included in
the equilibrium strategy were to drop out of the race it would not affect the equilibrium strategy. For
example, in Figure 7, the equilibrium strategy is Candidates A, B, and C all receive one third and D
received zero. If D were to drop out of the election, the equilibrium strategy would remain Candidates A,
B, and C each with one third.
Two Player Game Theory is a Condorcet voting method, that also has clone invariance, loser
independence, voter fairness, and all of the winners are relevant to the outcome. These fundamental
voting properties are not reflected with the majority of the voting methods currently implemented.
Two Player Game Theory provides winners that betters better represent the voter population and is
more difficult to manipulate than the current voting systems.
6
Wright, B. (2009, April 20). Objective Measures of Preferential Ballot Voting Systems.
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Works Referenced:
Bernholz, P. (1973). Logrolling, arrow paradox and cyclical majorities . Public Chocie, 15(1), 87-95.
Egelko, B. (2010, April 20). S.f. instant-runoff voting upheld. The Chronicle. Retrieved from
http://articles.sfgate.com/2010-04-20/bay-area/20856835_1_instant-runoff-voting-votingsystem-instant-runoffs
Electoral reform second choice or second-class? alternative ways of picking mayors are spreading.
(2011, October 22). The Economist, Retrieved from
http://www.economist.com/node/21533435/print
Quas, A. (2004). Anomalous outcomes in preferential voting. Stochatics and Dynamics, 4(1), 95-105.
Tabachnik, M. (2011, April 25). Strategy and Effectiveness: An Analysis of Preferential Ballot Voting
Methods.
Tideman, N. (1995). The single transferable vote. The Journal of Economic Perspectives, 9(1), 27-38.
Wright, B. (2009, April 20). Objective Measures of Preferential Ballot Voting Systems.