21 February 2011

Life Game

Game theory is a branch of applied mathematics that is used in many fields including the social sciences, economics, biology (evolution and ecology), engineering, political science, international relations, computer science, social psychology, philosophy (morality) and management. In game theory, there is a fundamental problem known as the prisoner's dilemma (PD). Here is an example of the PD:

Say, you and your new partner in burglary are captured near the scene of a burglary by the police. The police have insufficient evidence for a conviction. So, they separate you both in a different cell and offer the same deal. If you confess and your partner remains silent, you go free and your partner receives 10-year sentence, and vice-versa. If you both confess, you both receive 5-year sentence. If you both remain silent, you both receive 6-month sentence. What do you do?

If you both cooperate and remain silent, it is good for both of you. But, what if your partner defects you? In order to avoid the worst outcome, your rational choice is to defect.

PD can be summarized as below:

PD can be generalized as below:

PD can be used as a cost benefit analysis with points [benefit – cost]. Say, an animal A may help (with an expense of some cost) another animal B and hoping to get some help in return (benefit) from B in the future. But B may or may not help. In some cases, this can make a difference between life and death. For example, Vampire bats feed on blood at night. It is not easy for them to get a meal all the time, but if they do it is likely to be big one. Every day, some individuals will return completely empty, might starve to death. Often the lucky bats help the unlucky ones.

Let us give some points as an example (Note that the points only matter relative to each other, not the actual values). Say, the cost of help is 10 points, and the receiver gets the benefit of 15 points. If A helps B and gets no help from B in return, then A's total outcome: [benefit–cost = 0-10 = -10]. If A helps B and gets help from B in return, then A's total outcome: [benefit-cost = 15-10 = 5]. If A gets help from B and does not help B in return, then A's total outcome: [benefit-cost = 15-0 = 15]. This can be summarized as below:

If this game is played once, then A's rational choice (probabilistically safer bet) is to defect and not help B. But what if the game is played iteratively over and over again and A plays with not just B and many others at different times. Now, what is the successful strategy? American political scientist Robert Axelrod did a computer simulation for it. The idea is to take different strategies and convert them into computer programs; they play with each other continuously (this is something similar to what happens in animal population in evolutionary time). Many strategies were submitted by different institutions, scientists and mathematicians; some are simple and others are highly elaborate complex strategies. Though a successful strategy depends on the relative cost benefit points, other strategies in the population and many other factors, a simple robust strategy known as Tit-for-Tat is found to be a successful strategy. Tit-for-Tat strategy begins by cooperating on the first move and thereafter simply copies the previous move of the other player.

It can be analyzed why Tit-for-Tat is a successful strategy. Always cooperating strategy can be easily beaten by any defecting strategies in the population. Always-defecting strategy can be successful, if the population is filled with many cooperate strategies. Though Tit-for-Tat-except-first-defect strategy (first move defect and then behave like Tit-for-Tat) avoids losing from defecting strategies, but gains little with other carefully cooperating strategies like Tit-for-Tat. Tit-for-Tat strategy gains from other cooperating strategies and loses little from defecting strategies.

Axelrod describes many characteristics of Tit-for-Tat strategy. It is a 'nice' strategy, as it cooperates first. It is a 'forgiving' strategy, as it has a short memory for past misdeeds – only cares about last move of the other player. It is also 'not envious', as it does not strive for more points at others expense. But it is 'not a saint' strategy, as it does not cooperate unconditionally forever. Life is much more complex and involves many other factors. Yet, this gives some basic idea about how strategies like cooperation, helping others, being nice, forgiving and not envious might have been evolved in animals including humans.

This is our life game. Though we play this game with one another, we often fail to recognize it at bigger levels such as between groups, organizations, companies and nations. Unfortunately, most of the games (sports) we created are zero-sum games where one's gain is another one's loss.

Achilles: So, these evolutionary games become part of our behaviors. Before jumping to zero-sum games, why do we just play? Kids love to play... Why?

Tortoise: Playing is actually a serious business. It is evolved as an important learning strategy. That is how kids explore the world and test the water! That is how kids (humans and other animals) learn about the world (its characteristics, cause and effect, space and time, etc.), and what works and not.

Achilles: Why do adults play game?

Tortoise: We created many games as part of our further learning/training that were useful for our survival. Human’s early games are linked with using our survival tools such as using swords, throwing stones, bowing arrows, etc. The real practical values of games were slowly separated and disappeared. Now, most of the games stand on their own without any real practical values.

Achilles: But, games are good exercise and they promote healthy life style.

Tortoise: Games are good exercise, if we use them at right level. Most of the professional players are over doing it (harmful over exercise), if not using harmful drugs. Games are good exercise, if we promote everyone to play instead of just watching.

Achilles: It is entertaining to watch games. But I do think it is a good idea to have some non-zero-sum games and practically useful games.

Tortoise: In any case, we should just go out, play and enjoy our game and be sportive about it. Playing is a serious business and it is useful when it is playful!