Back in 2006, I took the undergraduate course Individual and Organizational Behaviour offered to students in the commerce programme at the University of Toronto. One of the highest quality lectures I have attended was given in this class by a guest speaker, a professor in the Rotman School of Management. During this lecture, students were put into groups and pitted against each other in a friendly competition. We faced off in what seemed to be a tragedy of the commons scenario and our gameplay was analysed by the speaker. However, I argue that what we were playing was not actually a tragedy of the commons scenario and that the analysis given in class was incorrect. Apparently, what I thought was extremely obvious was not. Or perhaps I’m just wrong; I’m no game theoretician.
In Part I of this blog post, I describe the game scenario. In Part II, I will describe the professor’s analysis followed by my take in Part III. In Part IV, I will ruminate about how this might apply to some contemporary social issues.
Note: I have forgotten some the details of the game such as the exact payoff ratios and the number of turns, but the mechanics described below are accurate enough for the purposes of discussion.
The Game Scenario
In the game, lasting 20 turns, each of four players starts off with no money. Each turn, players receive $100 and can choose to put $100 into a group pool (“cooperate”) or to keep it all to themselves (“defect”). Money in the pool accrues 200% interest at the end of the turn and is distributed equally to all players, including defectors. For example, if one player defects, the pool contains $900 (200% interest on $300 + the initial capital). The defector receives $100 + $225 (a quarter of the pool). Each other player receives $225. If only one person cooperates, the pool is $300 after interest (200% interest on $100 + the initial capital). Each of the three defectors receives $100 + $75. The cooperator, who contributed $100, only gets back $75.
This yields the following payoff matrix (change in money at the end of the turn):
At the end of the 20 turns, the professor converted each game dollar into a ticket for a draw for one of three prizes of real money. It matters little, but I think the amounts were one $100 prize and two $50 prizes. In any case, we will consider the optimal strategy to be the one which maximizes one’s chances of winning the real money.
During the class, each of the 100 or so student enrolled in the course was seated in front of a computer terminal and would face off against three other players for one game (20 rounds); only actions would be relayed to other players — no communication was permitted with the other players. We were told that the players were either students from our class or students receiving the same lecture at the University of Waterloo. ((At the end of the game, it was revealed that we faced other students in the room and computer “bots” with different pre-programmed playing strategies such as “always cooperate”. That is, there were no students at the University of Waterloo.))
Before we continue, what strategy do you think the professor proposed to be the optimal strategy? What do you think the optimal strategy is? ((You might like to read about the concept of superrationality that Douglas Hofstadter puts forth, though I can’t claim any knowledge of it. Having discussed my take immediately after lecture, both the professor and I assume the other players are rational, not superrational — the word didn’t even come up — though perhaps I’m too optimistic about the reasoning capabilities of the average rational human being.)) Stay tuned for part II of this 4-part series to hear the professor’s take.