Prisoner’s dilemma in long term supplier relationships

The games played by game theorist don’t exist in the real world. The assumption is that each player is motivated solely by their stated payoffs whereas in real life we are motivated to do more than maximise our material rewards. Sometimes we are motivated to maximise other’s rewards or the total rewards available to all players. Occasionally, we are motivated to minimise other’s rewards even at the expense of our own. But despite the inevitable simplification required by game theory, the very act of simplifying can uncover some useful features of the games we play.

Long term supplier relationships is the game I’m most interested in.

One of the simplest games (and therefore one of the most studied) is where two players choose to cooperate with each other or defect. Their respective payoffs depend on the choice made by the other player. Table A below is the standard way of representing the payoffs received by each player. The left-hand number in each cell is Player A’s payoff whilst the right-hand number is Player B’s payoff. For example the bottom centre cell (30,1) in the “Player A: Defect” row and “Player B: Cooperate” column shows that when Player A defects and Player B cooperates Player A gets $30 whilst Player B only gets $1. The efficiency of the player’s decisions is the sum of their payoffs relative the maximum sum of payoffs. In the above example the sum of the payoffs is $31 compared to a maximum payoff of $40 (calculated by adding the 20,20 values in the centre cell in the table).

The Prisoner’s Dilemma is a specific type of this game where the players when acting in their individual best interest create inefficient outcomes. The emotive example is a patrons exiting a theatre on fire. The surest method of saving the most patrons is for everyone to exit in an orderly manner. The surest method of saving yourself whilst everyone else is proceeding orderly is to rush the exit and get out first. Everyone rushes the exit. Many die. An inefficient outcome.

The following table demonstrates how this can occur. The “20, 20″ cell shows that, if Player A cooperates with Player B, they will each receive $20. The “1,30″ cell to the right shows that, if Player A cooperates and Player B defects, then Player A will receive $1 and Player B will receive $30. Likewise for the next row, the “30, 1″ cell shows that, if Player A defects and Player B cooperates, Player A will receive $30 and Player B will receive $1. The final cell shows that if both players defect then each will receive $5.

Table A: Standard prisoner’s dilemma

	Player B: Cooperate	Player B: Defect
Player A: Cooperate	20, 20	1, 30
Player A: Defect	30, 1	5, 5

Faced with these payoffs, what should Player A do? To answer this question, one need only look at the left-hand number in each cell. The “Player B: Cooperate” column shows that Player A will get $20 if he cooperates and $30 if he defects therefore, if Player B cooperates, Player A’s best choice is to defect. The “Player B: Defect” column shows that Player A will get $1 if he cooperates and $5 if he defects. Again, his best course of action is to defect. In game theory lingo, defecting is Player A’s dominant strategy. Note that defecting is also Player B’s dominant strategy. In game theory lingo, this is known as a Nash Equilibrium.

But if both Player A and Player B defect, they only receive $5 each - a total payoff of $10. Had they cooperated, they would have received $20 each for a total payoff of $40. This is the Dilemma.

The critical point to note is that if players do not communicate and are only playing a single round of the game then defecting is always the best option - you minimise your losses and potentially maximise your gains. However, if the game goes on for an infinite number of rounds then strategies other than always defecting result in higher payoffs.