Simulation for Prisoners’ Dilemma [PD]

Think of a game as a typical; game of poker where a sequence of moves is played, the game ends and then payoffs are realised. The sequence of poker-hands can be regarded as moves and they represent a game. Schooled players will look for connections between hands and each player will be observing what every other player is doing or not doing as the game unfolds.

The payoffs are determined by the components of the game. If the game is played just once then there is a unique payoff of (2,2). It can be any number but both players receive exactly the same amount. Each player has a dominant strategy and has the easiest of decisions faced with the PD matrix: cooperate and receive 2. Both players would prefer to be in the top left cell of the matrix, but because each player has a dominant strategy of competing, they will find themselves in the lower right cell of the matrix. A cartel between A and B might seem a solution, but with an inherent incentive to cheat, enforcement of the cooperative solution might prove to be difficult, vide the arguments in McNutt Law, Economics and Antitrust.

In four cycles of the game a player could receive 2+2+2+2 = 8. However, it is collectively that they each as players face a dilemma – how to obtain the cooperative outcome rather than the non cooperative outcome.

Read: (A,B): (0,3) Reads as 0 for A and 3 for B

	Player A
	cooperate
cooperate	2,2	0,3
compete	3,0	1,1

There is an element of trust involved; indeed, if one player believes that the other player will always cooperate then there is an incentive to cheat. For example, one player obtains 2 and expects to obtain 2 in the 2nd cycle of the game on the condition that the other player cooperates. Knowing that he will cooperate, the other player cheats and obtains a 3 [securing 2+3 instead of 2+2]. Once the cooperating player observes what has happened, he competes and both receive 1 in 3rd cycle and 1 in 4th cycle leaving the cheating player with (at a maximum) 2+3+1+1 =7 instead of 8. But the 3 may never materialise and he receives a 2+1+1+1 = 5.

Lesson: in trying to do better you can end up worse off! David Hume , writing in 18th century, captured the idea: ‘we can better satisfy our appetities in an oblique manner, than by their headlong and impetuous motion’

Simulation for Prisoners' Dilemma

Consider the case of small firm A v large firm B in a market setting where management have individual utilities or private benefits from competing in the market and in the game. In the latter case, by playing a game, management are players. This is an application of the management indifference relationship we had developed earlier in Kaelo. We ascribe a simple cost to each player.

The PD payoff matrix for this example is as follows: again players would prefer to be in the top left cell of the matrix, but because each player has a dominant strategy of competing, they will find themselves in the lower right cell of the matrix.

	Player A
	cooperate
cooperate	½ , ½	1/4, ¾
Player B
compete	¾, 1/4	0,0

The zero-sum assumption dictates that player A’s market share gain will be at the expense of the player B and vice vearsa. No two players move simultaneously. Both players prefer to avoid a zero-sum outcome represented by (0,0).

For the purposes of the game, assume that the private costs to player A of cooperating or ‘keeping his promises’ can be represented by c(A) = x2/2 and the private costs to player B can be represented by c(B) = 2x2. Neither player knows each others cost function.

The cost to player A of a cooperative outcome = 1/8 and the cost to player B = ½ suggesting that player B, given the specific cost function, is no better off. Management at B should attempt to reduce the private costs of playing the game. However if B is tempted to go for the ¾ payoff its costs would rise significantly to 1.11 with the inevitability of a (0,0) outcome as player A retaliates. So B with no legal sanction cooperates with A. Maximising private benefits can be difficult.

Play a PD game yourself at: www.serendip.brynmawr.edu/bb/pd.html

The application to management and business is very important particularly in a zero-sum market where two or three firms collectively have 100% of the market share. The strategy set is either to cooperate or compete. Strictly speaking, antitrust rules exist in many jurisdictions that act to dissuade firms from forming a credible cartel arrangement. Cartels are inherently unstable. Modern companies can compete by cooperating through joint ventures and outsourcing. One interesting hypothesis is to consider the presence of an acute zero-sum constraint in an oligopoly market (with 5 players) as a trigger for a merger wave in the industry.

Polar Bear and Seal Dilemma

As the ice cap melts, there is less ice for the seals to rest. Less resting places makes it harder for the polar bears to capture the seals as they spend more time swimming in the waters. Polar bears cannot swim as well as seals. So do polar bears spend more time on fewer ice caps waiting for the seals to rest or do they improve their ability to swim longer in pursuit of the bears? For the seals, knowing that polar bears cannot swim as well and as long as they can, the optimal action is to stay longer in the water, but they risk capture by the polar bear, who, knowing that the seals believe that polar bears cannot swim as well as they, adapts to swim better, pursues the seals until they rest on an ice cap in the belief that there will be no bears on the presumption that they cannot swim as well as they.

However, if more seals stay longer in the ice water, polar bears unable to swim as well, will fall in population as they loose a vital food source. And as the ice caps continue to melt, fewer seals will survive. Fewer seals mean fewer polar bears, so as the ice caps melt it is a no win outcome for both the polar bears and the seals. Correct? A Darwinian outcome could evolve, but that depends on nature. In a business context, is there a lesson from the polar bears and seals dilemma? It may be is that as a market evolves the player who adapts better to an evolving market has a greater probability of surviving in that market.