Cooperative Games

When you tell someone you were playing a game, even if the person has never heard of the game you were playing, there is a stock question they can use to avoid seeming ignorant: “did you win?” Except not – and that is the topic of today’s Occupy Math. Cooperative games are games where the players are working together to achieve some goal – depriving the more competitive among us of the joy of beating others, but also potentially helping develop important life skills.

Competition is a great motivator – but cooperation is one of the great strengths of the human species.

Occupy Math has spent a lot of time teaching software agents to play the game prisoner’s dilemma which is used as a mathematical model of cooperation and conflict. The situation that it models is two prisoners, placed in different rooms by the police, and offered a deal if they testify against their accomplice. In this case cooperation consists of maintaining silence and supporting your accomplice while defection consists of testifying against your accomplice. The game is used to mathematically investigate under what circumstances individuals will choose to cooperate or engage in conflict. Extending the game beyond its roots in modeling a single crime, it is often played for several rounds.

In the book The Evolution of Cooperation, by Robert Axelrod, the prisoner’s dilemma model is used to explain a Christmas Day soccer match, played in no-man’s land, between the opposing forces during World War I. The book also reports an excellent strategy that has entered the language of international diplomacy: tit-for-tat. This strategy never defects first – rather it only defects when its opponent has just defected. This strategy elicits cooperation, punishes defection, and is easy to understand. Tit-for-tat won two tournaments for prisoner’s dilemma strategies run by Axelrod. The book is readable without needing to know much math and it is not long – but it conveys some fundamental points about cooperation and conflict.

Prisoner’s Dilemma is an early example of a game where cooperation is possible

At this point we introduce a game for 4-12 year olds that teaches a fundamental mathematical skill in addition to helping train the players in cooperation. A key skill in math is to explain something with great clarity to another person. The proof is the core of advanced math – this requires you to explain a claim so clearly that it becomes obvious. One of my favorite authors Lois McMaster Bujold had the protagonist of Ethan of Athos think to himself:

“It all fit, with the overpowering self-evidence of a mathematical proof.”

The game is for two players and requires two identical collections of duplo blocks (or other, similar construction toys)
and a screen that sits between two players and prevents them from seeing the other’s construction. One player builds a structure with the blocks. He then gives directions to the other player that permit them to duplicate the construction. The instructions need to avoid feedback like “no, not like that!” – remember you cannot see the structure the other player is building.

The instructions should consists of statements like “put the red square block in the middle of the longer blue block”. The goal is to learn to describe the construction process concisely and correctly. This requires a very high level of cooperation between the players. Each game should consists of two rounds in which each player gets a turn as the instructor. This game builds the ability of precise description and explanation that is the foundation of mathematical proof. This game also teaches and enhances language skills.

There are also many cooperative games of a more traditional type.

A list of cooperative games is maintained by Kadon games. Kadon is also an excellent source of games and puzzles, including cooperative games. One of the best known cooperative games is Pandemic in which players cooperate to head off the emergence of a global pandemic. Cooperative games like Pandemic build collaborative problem-solving skills. It is also a fun game to play.

Another genre, the semi-cooperative game, exists. Games like The Resistance, (goal: revolution) Werewolf, (goal: figure out who is the werewolf) and Camelot (goal: strengthen the round table) are games in which the players are cooperating to achieve a goal except that one or two players are working against that goal. While the villagers in Werewolf are trying to find the werewolf, for example, the werewolf is eating villagers. These games manage to hone cooperative skills while leaving an element of competition in the game.

The game Scotland Yard is a wonderful blend of cooperation and conflict. All but one of the players are constables with limited travel passes roaring about a map of London trying to apprehend the despicable Mr X, a miscreant with unlimited travel passes. This game teaches planning, resource management, and connectivity as well as requiring cooperation among the constables. Over-competitive constables who try too hard to be the one to make the collar are Mr. X’s best friends.

Cooperative games are becoming more common, as board games, as training tools, as online games, and in the kind of games used to do political and economic planning. Occupy Math would love to hear about cooperative games that you’ve encountered. Be sure to rate their quality as games and let us know if there were players that managed to be competitive in spite of the cooperative nature of the game.

I hope to see you here again
Daniel Ashlock
University of Guelph
Department of Mathematics and Statistics


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