Cooperation and Power

Mathematics majors Andy Clemens and Peter Schrock worked with David Housman on Maple Scholars projects examining the evolution of cooperation among self-interested individuals and measuring power in weighted voting systems.

Maple Scholars is an eight-week summer program that gives students the opportunity to participate in independent research projects alongside Goshen College faculty of various disciplines.   Scholars keep each other informed of their progress in weekly seminars, and the program ends with a celebration involving a public poster session, presentations, and a dinner.  Andy and Peter also presented his work during a visit to the Valparaiso Experience in Research by Undergraduate Mathematicians, the 2013 Indiana Undergraduate Mathematics Research Conference held at Indiana University Bloomington, and the 2013 MathFest held in Hartford, Connecticut.

Andy Clemens presenting his work at MathFest.

Andy Clemens presenting his work at MathFest.

The world is full of many different kinds of conflict, which vary in many ways.  A common type of conflict occurs when two or more participants have their own best interests in mind, but must work together, at a cost, to achieve those interests.  In game theory, this is known as the prisoner’s dilemma.  The most basic form of this is when two players can either cooperate, which would help the opponent but hurt the cooperator by a smaller amount, or defect, which has no effect on either player.  No matter what the other player does, each player always does better by defecting; however, each player does better when both cooperate than when both defect.  This dilemma can be resolved with both players choosing to cooperate when the game is played in multiple rounds.  Andy Clemens examined what happens when communication errors may occur and players can adopt strategies with one or more periods of memory.  Stable strategies were found using a process of evolution: different strategies play each other, the weaker ones die off, and the stronger ones remain and possibly “mutate” into new strategies, until the strongest strategies dominate over all the others.

Peter Schrock at MathFest

Peter Schrock presenting his work at MathFest.

In a nine person committee where every member has one vote and a majority, or five votes, is required to reach a consensus, five members would stand to increase their power by agreeing to form a bloc. By secretly voting before meeting as a whole committee, the five could come to a consensus by majority and then all agree to vote the same way in front of the full committee. Because the bloc effectively controls the committee, each member would have one fifth of the full committee’s power, instead of the one ninth they would receive without the bloc. Three particularly Machiavellian members of the five could then theoretically form a bloc within the bloc of five and control the whole committee. If the full committee required unanimity instead of a majority, forming blocs that require a majority decreases the power of individual voters. Members of such a bloc would lose their ability to single handedly veto a proposal and thus would lose power.  Peter Schrock defined a new way to measure power in a voting system by taking into account the possibility of forming blocs.

Final reports can be accessed from David Housman’s Undergraduate Research Page.