This rather long post is a journey into a fascinating and fairly recent area of mathematics, “the game theory”. I must say that my knowledge in this area is limited to an introductory book and a few articles here and there, and I don’t claim to be an expert in any way. But if you like mathematics as much as I do, you will probably enjoy this post. If you find any errors, I appreciate you letting me know, so I can correct them.
Introduction
The theory of games was created in 1928 by one of the true geniuses of mathematics, John Von Neumann with the publication of his min-max theorem. Later, he and Oskar Morgenstern expanded the initial ideas and showed their application in economics in a paper they published in 1944. From then on, this theory found its way into many areas of science, from sociology, psychology, political science, and economics to the science of evolution and even ecology.
We can call the game theory the theory of human behavior or the theory of making decisions. In a game, the players are faced with different choices and must decide among several strategies to maximize their winnings. Game theory offers a model and a mathematical system to compare different strategies to try to find the possible optimum ones and predict the eventual outcome. Many of the real life situations can be modeled with this theory. This area is also filled with paradoxes which are as complex as they are fascinating.
The Two-Player Zero-Sum Game
The simplest form of a game which can be used to illustrate the basic ideas of game theory is a game played by two people where the winning of one player results in the lose for the other player by the same amount and vice versa. An example of this game is the election in a two party system where the votes earned by one candidate are lost by the other candidate. We consider the US elections as an example, and Bush and Kerry as the two players. For simplicity, we consider two choices for each candidate. Bush has the choice of staying with his right-wing policies or lean to the center. Kerry has the choice of adopting more liberal policies or lean to the center. In a zero-sum game such as this, we consider only the percentage of votes for one of the candidates, say Bush, as the outcome of the game, because the winning of the other candidate can be deduced from that.
Let’s say the opinion polls show that if both candidates stay moderate, the votes are dead even. Both have a core of supporters that they can always count on no matter what. Bush can mobilize the religious right if he leans toward right-wing policies, but there are a smaller group of moderate republicans whom he will lose to Kerry as a result of his right leaning agenda. Kerry can also mobilize progressives if he articulate liberal policies but he will also lose some moderate democrats to Bush if he chooses to do so. So, based on the opinion polls a 2x2 matrix of election results is constructed.
Election Game
| Kerry
|
|---|
| Liberal | Moderate
|
|---|
| Bush | Conservative | 54 | 52
|
|---|
| Moderate | 48 | 50
|
|---|
Moderate-Moderate entry is 50% showing equally divided votes. Bush going Right, he picks up religious right in large numbers but loses some moderates to Kerry, but he gains more than he loses, so we arrive at 52% number. If Kerry moves to Liberal side while Bush stays moderate, progressives join Kerry, boosting his chances. He loses some moderates, but overall he gains which reduces Bush’s vote to 48%. If both Kerry moves to Left and Bush moves to Right, Bush does even better than 52% because the net effect of moderates switching sides is to his favor due to Kerry’s liberal stands, and Kerry’s progressives are not enough to compensate for the gain in religious right votes, so we arrive at 54%.
A quick look at the table shows where the final result of the election will be. Bush turning right is his dominant strategy and that will rob Kerry of his only chance of winning which is the box with number 48. So, Kerry has to stay moderate to avoid the crushing defeat spelled out by number 54. So, the final result of the election is 52% for Bush, and the final choices of the players are Right for Bush and Moderate for Kerry if both act reasonably.
Notice that 52 is the minimum of its row and maximum of its column. So, the outcome of this game is called the “minmax” value of the game and the optimum strategy of the players is the minmax strategy of the game. Any deviation from the minmax strategy for any player, assuming his opponent is playing his minmax strategy, will result in a less desirable outcome for that player (look at the table and try to convince yourself of this).
This game, where the dominant strategies of the players are clear, is simple, and predicting the outcome is no big deal. But not all the zero-sum games are that simple and most require a mixed strategy where the minmax strategy is a combination of two or more decisions taken with certain probabilities.
So, let’s look at a more complex two-player zero-sum game, the escape game. A convict escapes the prison and a marshal goes after him. There are two routes for escape, the road or the woods. If both take the road, the convict will be captured. If both go to woods, there is a 50% chance of escape. If one goes to the woods, and the other to the road, the convict will escape. So, we can construct the following table.
Escape Game
| Convict
|
|---|
| Road | Woods
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|---|
| Marshal | Road | 0 | 1
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|---|
| Woods | 1 | 0.5
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|---|
This game does not have a pure strategy. At first, it seems that the convict should take the woods because at least he has a chance to escape, and the marshal needs to go after him. So, the outcome of the game must be 0.5 which is the probability of the escape through woods. But if this is the best strategy, then the convict can assume that the marshal is definitely going to the woods to catch him, so he can take the road and escape (value=1). But wait, marshal can also make the same argument and block the road and catch him. This circular argument can go on forever.
This game requires a mixed strategy and the outcome of the game is the average winning of the players considering the probability of each decision in that mixed strategy. In the escape game, if the convict can play the game multiple times, he should take the road 1/3 of the time and go through the woods 2/3 of the time to maximize its chances of winning. If you calculate its probability of escape based on these probability values, you arrive at 2/3 no matter what decision the marshal makes (Try to calculate this and convince yourself that this is true). So, the convict can achieve better odds of escape than using pure strategy of going through the woods all the time. Marshal also needs to use his minmax strategy which is to block the road 1/3 of the time and go to the woods 2/3 of the time to guarantee that the convict will not achieve more than 2/3 chance for his escape.
The existence of this minmax strategy does not mean that there is no way the convict can achieve better results. If they both act unreasonably and do not follow their minmax strategy, it is possible to come up with a better outcome for the convict or for the marshal. For example, if the marshal takes a 50/50 strategy, and the convict always goes through the woods, the outcome is .75 (why?) which is more than 2/3.
The fundamental theorem which started the game theory was the minmax theorem that John Von Neumann published in 1928. The theorem states that for every two-player zero-sum game, there exists a value V which is the average winning that a player is expected to get, if players play reasonably. This value is the minmax value of the game and the strategy used to achieve it is the minmax strategy.
The Two-Player Non-Zero-Sum Game
Zero-sum games are not very common in real life. The non-zero-sum games have more real life applications. They are also much more complex, difficult to predict the outcome, and more interesting. In these games, the minmax theorem does not always hold true anymore. So, the outcome of the game and the optimum strategy does not always depend only on the values in the table.
Prisoner’s Dilemma
The most famous example of this type of the game which shows the difficulty associated with these problems is the so-called “prisoner’s dilemma” game. In this game two players commit a crime and are caught. But there is not enough strong evidence against them, so if they stay quiet and do not admit to anything, they both only get 1 year sentence. So, the district attorney separates them and offers each one a deal that if one of them confesses and testifies against his partner, and his partner does not confess, he can go free and his partner gets 15 years in prison. But if they both confess they both will end up with a 5 year sentence. So, we arrive at the following table:
Prisoner’s Dilemma Game
| 2nd prisoner
|
|---|
| Confess | Remain Silent
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|---|
| 1st prisoner | Confess | (-5,-5) | (0,-15)
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|---|
| Remain Silent | (-15,0) | (-1,-1)
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|---|
The negative sign indicates lose. The so-called reasonable strategy for the players is to maximize their gains, or in this case minimize their lose. At the first glance, it seems that the best strategy is to stay quiet so they get only 1 year sentence. But, if this is the best strategy, the 1st player can predict that and choose to confess and go free. The other player can also make the same argument and confess and they both end up with 5 year in prison.
The difficulty in this game arises from the fact that if one does not consider the other player’s gain, the best strategy for both players is to confess in order to minimize their own lose. If you construct the table with only the winning of one player and analyze it the same way as you would analyze the zero-sum game, such as the election game that we discussed before, you can clearly see that the confession-confession box is the minmax point in the table and is the dominant strategy for each player (do this and convince yourself that this is the case).
In the zero-sum game, if both players worked toward maximizing their gain, there was a guarantee that they arrive at an optimum solution for both according to the minmax theorem. But as you can see in this case, this is no longer true. In this case, they both can get better results if they both choose a different strategy and cooperate.
In many experiments done where people played this or similar games, surprisingly they played the non-cooperative strategy more than cooperative strategy. However, this result does not necessarily mean that people will behave the same way in real life. For example, if the values in the table are changed, so that the numbers are positive corresponding to gains, the objective of the game becomes maximizing gains rather than minimizing lose. The two tables should be mathematically identical, but the results change. The players tend to use cooperative strategy more. Although, the non-cooperative strategy still wins, the margin is narrower.
Another Game
| 2nd player
|
|---|
| Don't cooperate | Cooperate
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|---|
| 1st player | Don't cooperate | (1,1) | (8,0)
|
|---|
| Cooperate | (0,8) | (5,5)
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|---|
Experimenters also found many other factors such as communication between the players, history of the plays, the value of payoffs, etc affects the outcome.
Capitalism vs Socialism
From the studies on the “prisoner’s dilemma” games, some people concluded that selfishness is part of the human nature and even if most people adopt the cooperative strategy in their life, the few, who don’t, accumulate gains to eventually dominate the rest. They can give the failed socialist experiments of the 20th century as examples, where the cooperative strategy did not work and privileged few finally took advantage of the rest and dominated them for their own gains.
There are many other experiments with these games that contradict the conclusions about inherent selfishness, but let’s approach the comparison from a different point of view. Let’s see if we can justify changing the numbers in the table to find out if there is a way to resolve this paradox and arrive at a minmax solution for the society. Let’s see under what condition the cooperative strategy or inversely a non-cooperative strategy can dominate.
The defenders of capitalism can claim that the table of the prisoner’s dilemma game does not reflect the reality of the capitalism because the complete non-cooperative outcome is not the objective of the capitalist society. They can try to resolve the conflict which arises in the prisoner’s dilemma game by proposing the following table for the gains.
Capitalist’s Game
| Worker
|
|---|
| Be Greedy | Cooperate
|
|---|
| Capitalist | Get Incentive | (2,2) | (8,7)
|
|---|
| Give in | (3,6) | (5,5)
|
|---|
Take a moment to study the table and see what you think the outcome of the game should be and what strategies are optimums for each player with the proposed values.
Here is how the argument of the defenders of capitalism goes. They claim that for progress in science and technology, etc people need incentives. So, the fist row in the table is the incentive row not greed. That is what motivates the capitalist to find better ways of doing things, reduce cost and maximize profit. Of course, the capitalist needs the cooperation of the workers to turn the wheels in the factories and achieve this result, so if they work together like this, they can maximize the overall gain which is, let’s say, 15 units and they can both share that unequally as (8,7). The capitalist should get a bigger share, of course, so he has the incentive to keep this going.
If the workers get greedy, do not cooperate, form unions, and demand more in terms of better wages, health care, etc, then the capitalist has two choices. If he does not give in to their demand, the workers may go on strikes, the production stops, and the total output will be reduced to 4 units that they can get equal shares and end up with (2,2) result.
If the capitalist give in to workers demand and provide better wages, health care, etc, some of the businesses will lose money, some go bankrupt, the production stops and overall we end up with only 9 units. The benevolent capitalist will give up some of his share to the greedy worker and we end up with (3,6).
If both the capitalist and the worker cooperate, the capitalist gets a bigger share than 3, improves things and the overall production becomes better. But it will never reach the optimum level, because the capitalist does not have enough incentive. They overall get 10 units which they divide equally at (5,5).
From this table, it becomes obvious that in the capitalist’s paradise, the best solution that gives both parties the best result is to keep the inequality, give incentive to the capitalist, and make sure the workers cooperate and everyone will be better off.
On the other hand the defenders of mutual cooperation write a different scenario. They change the table as follows:
Socialist’s Game
| Worker
|
|---|
| Be Greedy | Cooperate
|
|---|
| Capitalist | Be Greedy | (2,2) | (5,4)
|
|---|
| Cooperate | (4,5) | (6,6)
|
|---|
In this table the cooperation strategy gives the best results for both sides and it will be the optimum and stable outcome of the game. Their argument to justify the number in the table goes something like this:
The profits of each production activity should not only take into account the direct costs associated with that activity such as raw material, buildings, infrastructure, utilities, etc, but also should take into account other costs that is forced upon the whole society. These costs include the environmental damage that the activity causes, the social services that the society has to provide for each and every member of the society such as food, shelter, health care, security, etc which is not universally provided in the capitalist society. Let’s say the overall maximum gain is less than maximum 15 which was achieved in the Capitalist’s game due to other factors such as reduced incentives, etc, and let’s say 14 units is gained. The costs add up to about 2 units and the 12 remaining units gets divided equally between the two players and we arrive at (6,6) number for the cooperative outcome.
If from this point, we move to the greed row, the production may go up, but the cost to the society goes up much more. Since the personal gain is the objective of the greedy capitalist, the environmental concerns goes out the window and the cost to the planet goes up in terms of global warming, pollution of the ground waters, destruction of rain forest, etc. Since under the capitalist society, there is no incentive to provide universal protection for the members of the society, rates of crimes and disease will go up which increased the overall cost to the society. Another cost is the cost of the inefficiencies of the market, the over-production that leads to recessions, etc. Also, greed and competition at international arena lead to war and destructions and its burden is put on the human society as a whole and should be subtracted from the overall gain. So, we will generously arrive at 9 units which can be divided between the two players (5,4)
If the roles are reversed, we get the same results, but the distribution will be different, so we get (4,5).
With both players playing greedy, we get the same result as in the capitalist’s game and end up with (2,2).
Therefore, in this table the best outcome is (6,6) where cooperation between the members of the society will reduce the overall cost to the planet and helps the standard of the living of all people in the world.
This, they argue, is why progressive forces fight for environment causes. They work toward achieving equality and human rights. They struggle against poverty and war. These struggles will show the true cost of greed and will eventually shift people’s perception from the capitalist game to a cooperative game where everyone is a winner.
Which game plan do you follow?
