The Prisoner's Dilemma

To be able to understand the strategies some games are played according to it is necessary first to get the idea of the prisoner’s dilemma.

Two burglars, Bob and Al, are captured near the scene of a burglary and are given the "third degree" separately by the police. Each has to choose whether or not to confess and implicate the other. If neither man confesses, then both will serve one year on a charge of carrying a concealed weapon. If each confesses and implicates the other, both will go to prison for 10 years. However, if one burglar confesses and implicates the other, and the other burglar does not confess, the one who has collaborated with the police will go free, while the other burglar will go to prison for 20 years on the maximum charge.

The strategies in this case are: confess or don't confess. The payoffs (penalties, actually) are the sentences served. We can express all this compactly in a "payoff table" of a kind that has become pretty standard in game theory. Here is the payoff table for the Prisoners' Dilemma game:

 

 

		Al
		confess	don't
Bob	confess	10,10	0,20
	don't	20,0	1,1

 

The table is read like this: Each prisoner chooses one of the two strategies. In effect, Al chooses a column and Bob chooses a row. The two numbers in each cell tell the outcomes for the two prisoners when the corresponding pair of strategies is chosen. The number to the left of the comma tells the payoff to the person who chooses the rows (Bob) while the number to the right of the column tells the payoff to the person who chooses the columns (Al). Thus (reading down the first column) if they both confess, each gets 10 years, but if Al confesses and Bob does not, Bob gets 20 and Al goes free.

So: how to solve this game? What strategies are "rational" if both men want to minimize the time they spend in jail? Al might reason as follows: "Two things can happen: Bob can confess or Bob can keep quiet. Suppose Bob confesses. Then I get 20 years if I don't confess, 10 years if I do, so in that case it is best to confess. On the other hand, if Bob doesn't confess, and I don't either, I get a year; but in that case, if I confess I can go free. Either way, it is best if I confess. Therefore, I'll confess."

But Bob can and presumably will reason in the same way – so that they both confess and go to prison for 10 years each. Yet, if they had acted "irrationally," and kept quiet, they each could have gotten off with one year each.

What has happened here is that the two prisoners have fallen into something called a "dominant strategy equilibrium."

· The Prisoners' Dilemma is a two-person game, but many of the applications of the idea are really many-person interactions.

· We have assumed that there is no communication between the two prisoners. If they could communicate and commit themselves to coordinated strategies, we would expect a quite different outcome.

· In the Prisoners' Dilemma, the two prisoners interact only once. Repetition of the interactions might lead to quite different results.

· Compelling as the reasoning that leads to the dominant strategy equilibrium may be, it is not the only way this problem might be reasoned out. Perhaps it is not really the most rational answer after all.

Assignments:

1. Active vocabulary:

To confess, to implicate, to collaborate, payoff, penalty, interaction, to assume.

2. Find synonyms among the following words:

To cooperate, the most influential, penalty, to commit, to guide, to assume, prison, to lead, dominant, to accomplish, jail, to suppose, punishment, to interact

Read the dictionary definitions and find the defined words in the text.

1. Situation in which one has to choose between two things.

2. A person who breaks into a house at night in order to steal.

3. Punishment for wrong-doing, for failure to obey rules or keep an agreement.

4. Something the most important or influential.

5. Department of government, body of men, concerned with the keeping of public order.

Turn the sentences into the Active Voice.

1. Bob and Al are captured near the scene of a burglary and are given the "third degree" separately by the police.

2. Some games are played according to certain rules.

3. The two numbers in each cell tell the outcomes for the two prisoners when the corresponding pair of strategies is chosen.

4. If each confesses and implicates the other, both will be sentences for 10 years.

5. The table is read like this…

Fill in the gaps with the words and expressions from the text.

1. Two burglars, Bob and Al _____ near the scene of a burglary and are given the "third degree" separately by the police.

2. If neither man _____, then both will serve one year on a charge of carrying a concealed weapon.

3. The one who has _____ with the police will go free.

4. The Prisoners' Dilemma is a two-person game, but many of the applications of the idea are really many-person _____.

5. The two prisoners have fallen into something called a “_____ strategy equilibrium.”

Text 5

Strategies

For the game of coin tossing:there are two pure strategies: play heads or tails. For stone-paper-scissors there are three pure strategies: play stone or paper or scissors. In both instances one cannot just continually play a pure strategy like heads or stone because the opponent will soon catch on and play the associated winning strategy. What to do? There are some ways to control how to randomize. For example, for stone-paper-scissors one can toss a six-sided die and decide to select stone half the time (the numbers 1, 2 or 3 are tossed), select paper one third of the time (the numbers 4 or 5 are tossed) or select scissors one sixth of the time (the number 6 is tossed). Doing so would tend to hide one’s choice from the opponent.

For two-person zero-sum games, the 20th century’s most famous mathematician, John von Neumann, proved that all such games have optimal strategies for both players, with an associated expected value of the game. Here the optimal strategy, given that the game is being played many times, is a specialized random mix of the individual pure strategies. The value of the game, denoted by v, is the value that a player, say Player 1, is guaranteed to at least win if he sticks to the designated optimal mix of strategies no matter what mix of strategies Player 2 uses. Similarly, Player 2 is guaranteed not to lose more than v if he sticks to the designated optimal mix of strategies no matter what mix of strategies Player 1 uses. If v is a positive amount, then Player 1 can expect to win that amount, averaged out over many plays, and Player 2 can expect to lose that amount. The opposite is the case if v is a negative amount. Such a game is said to be fair if v = 0. That is, both players can expect to win 0 over a long run of plays. The mathematical description of a zero-sum two-person game is not difficult to construct, and determining the optimal strategies and the value of the game is computationally straightforward. It can be shown that heads-tails is a fair game and that both players have the same optimal mix of strategies that randomizes the selection of heads or tails 50 percent of the time for each. Stone-paper-scissors is also a fair game and both players have optimal strategies that employ each choice one third of the time.

Assignments:

1. Active vocabulary:

Associated winning strategy, to randomize, to toss, designated optimal mix of strategies, computationally.

2. Arrange the following words according to the parts of speech they belong to:

Instance, individual, guarantee, strategy, expect, amount, determine, selection, optimal, randomize, description, value, specialized, averaged, computationally.

3. Give the English equivalents of:

Чистая стратегия, игра с ненулевой суммой, безразлично (неважно), подбрасывать игральную кость, честная игра, «камень-ножницы-бумага».