Game theory is a distinct and interdisciplinary approach to the study of human behavior. The disciplines most involved in game theory are mathematics, economics and the other social and behavioral sciences. Game theory (like computational theory and so many other contributions) was founded by the great mathematician John von Neumann. The first important book was The Theory of Games and Economic Behavior, which von Neumann wrote in collaboration with the great mathematical economist, Oscar Morgenstern. Certainly Morgenstern brought ideas from neoclassical economics into the partnership, but von Neumann, too, was well aware of them and had made other contributions to neoclassical economics.
Assignments:
1. Answer the questions:
1) What is game theory?
2) What disciplines most involved in game theory and why?
3) Who can be called a founder of game theory?
2. Give a brief summary of the article using the following words and expressions: a branch of applied mathematics, an interplay between parties, the study of human behavior, to be founded by, in collaboration with, to make contribution.
Text 2
What is a game?
Game: A competitive activity involving skill, chance, or endurance on the part of two or more persons who play according to a set of rules, usually for their own amusement or for that of spectators (The Random House Dictionary of the English Language, 1967).
A game is the set of rules that describe it. An instance of the game from beginning to end is known as a play of the game. And a pure strategy is an overall plan specifying moves to be taken in all eventualities that can arise in a play of the game. A game is said to have perfect information if, throughout its play, all the rules, possible choices, and past history of play by any player are known to all participants. Games like tick-tack-toe, backgammon and chess are games with perfect information and such games are solved by pure strategies. But whereas one may be able to describe all such pure strategies for tick-tack-toe, it is not possible to do so for chess, hence the latter's age-old intrigue.
Games without perfect information, such as matching pennies, stone-paper-scissors or poker offer the players a challenge because there is no pure strategy that ensures a win. Games such as heads-tails and stone-paper-scissors are also called two-person zero-sum games. Zero-sum means that any money Player 1 wins (or loses) is exactly the same amount of money that Player 2 loses (or wins). That is, no money is created or lost by playing the game. Most parlor games are many-person zero-sum games. Not all zero-sum games are fair, although most two-person zero-sum parlor games are fair games. So why do people then play them? They are fun, everyone likes the competition, and, since the games are usually played for a short period of time, the average winnings could be different than 0.
Assignments:
1. Active vocabulary:
A competitive activity, a set of rules, an instance of the game, a play of the game, a pure strategy, eventuality, a game with (without) perfect information, a zero-sum game, tick-tack-toe, backgammon, chess, stone-paper-scissors, matching pennies.
2. Give the definitions of the following notions in English:
A game, a play of the game, a pure strategy, a game with perfect information, a game without perfect information, a zero-sum game, a fair game.
3. Fill in the gaps with the words and expressions from the text.
1. Game is a competitive activity involving skill, chance, or endurance on the part of two or more persons who play according to ________.
2. An instance of the game from beginning to end is known as ______.
3. A pure strategy is an overall plan specifying moves to be taken in all ______ that can arise in a play of the game.
4. A game is said to have ______if, throughout its play, all the rules, possible choices, and past history of play by any player are known to all participants.
5. Games like _______, _______ and _____ are games with perfect information and such games are solved by pure strategies.
6. Games such as ______ and _____ are called two-person zero-sum games.
7. Zero-sum means that any money Player 1 wins (or loses) is exactly the same amount of money that Player 2 ___(___).
4. Match the notions with the definitions.
| A game | No money is created or lost by playing this kind of a game. |
| A pure strategy | A game that has got no pure strategy that ensures a win. |
| A zero-sum game | A game, all the rules, possible choices, and past history of its play are known to all participants. |
| A play of the game | A game played by two persons only. |
| A two-person game | A competitive activity involving skill, chance, or endurance on the part of two or more persons. |
| A game with perfect information | An instance of the game from beginning to end. |
| A game without perfect information | An overall plan specifying moves to be taken in all eventualities that can arise in a play of the game. |
5. Match the names of the suits with the pictures.
Hearts, spades, diamonds, clubs.
1) ♠ 2) ♣ 3) ♥ 4) ♦
6. Answer the questions:
1) What is a game?
2) What do we call a play of the game?
3) What is a pure strategy?
4) What games with perfect information can you name? Why?
5) Have games without perfect information got pure strategy?
6) What are zero-sum games?
7. Characterize each game (chess, tick-tack-toe, backgammon, stone-paper-scissors, matching pennies) according to the following criteria:
A game with (without) perfect information, a zero-sum game, a two-person (many person) game
Finish the sentences suggesting your own ideas. Remember, that after ‘because of’ we use either a noun/pronoun or an –ing form.
1. Chess is a two-person game, because of…
2. Stone-paper-scissors is a game without perfect information, because…
3. Matching pennies is not a game with perfect information, because …
4. Stone-paper-scissors is a zero-sum game, because of …
5. Tick-tack-toe is a game with perfect information, because …
Text 3
Read the rules of two games and translate them into Russian.
How to play some games
A Game of Coin Tossing: Two players take any coin at hand. Big-size coins are more fun. One of them starts tossing the coin and continues for a long time. It will be convenient to assume that the tosses occur at equal intervals of time. If heads, Player 1 wins, say, one kopeck, if tails, Player 2 wins and his opponent pays.
Rock (Stone), Scissors, Paper: Two players show with their hands one of three figures: rock (that’s a fist), scissors (two fingers are shown), or paper (a player shows a palm); actually they are to choose randomly among these three options, with equal weights. Rock is stronger than scissors, scissors are stronger than paper and paper in stronger than rock. The one, who shows the stronger figure, wins. The fun of playing this game comes from trying to guess and exploit the other player's choices.
Assignments:
Pair work: think of some game you know very well, describe its rules to your neighbour, by analogy with the above mentioned games. Let your neighbour guess which game you were speaking about.
Text 4
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."
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
3. 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.
4. 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…
5. 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».
4. Insert one of the following prepositions that will best suit the context: because, because of, in spite of, despite.
1. In both instances one cannot just continually play a pure strategy like heads or stone ____ the opponent will soon catch on and play the associated winning strategy.
2. For two-person zero-sum games, the 20th century’s most famous mathematician, John von Neumann, proved that just ____ all such games have optimal strategies for both players they can be called fair games.
3. For example, for stone-paper-scissors one can toss a six-sided die and ____ the numbers tossed select either stone, paper or scissors.