Strategy games fb2. Avinash Dixit - Game Theory
Title: Game Theory. The Art of Strategic Thinking in Business and Life
Writer: Barry J. Nalebuff, Avinash Dixit
Year: 2008
Publisher: Mann, Ivanov and Ferber (MYTH)
Genres: Personal growth, Foreign psychology, Foreign educational literature, Popular about business
About the book “Game Theory. The Art of Strategic Thinking in Business and Life" Barry J. Nalebuff, Avinash Dixit
Many people know that our life is a game. But are you familiar with the rules of this game, with some tricks that help you win and predict your opponent’s moves? Barry J. Nalebuff and Avinash Kalamakar Dixit shared the secrets of successful gaming activities that can be applied both at work and in everyday life.
Book “Game Theory. The Art of Strategic Thinking” is a real tutorial on how to survive in difficult modern life with all sorts of intricacies and pitfalls. The authors explain in accessible words the essence of modern society - everyone plays in it. And although life's games are distinguished by a greater branching of moves, their nature is the same as in simple children's games. Barry J. Nalebuff and Avinash Kalamakar Dixit give many examples from various spheres of life - show business, sports, politics, everyday life, family life, business activities, etc. — showing people's involvement in strategic games.
The main idea of the book is that in order to play successfully, you need to think correctly and plan everything clearly. This is impossible without basic knowledge of game theory. You should also remember your school knowledge of algebra, master the laws of logic and some economic fundamentals. In this fundamental work, everything has been analyzed very carefully, so it will not be possible to “swallow” the book superficially - each chapter needs to be thought through and systematized. However, despite the academic nature of the work, Barry J. Nalebuff and Avinash Kalamakar Dixit wrote it in an accessible language, following a clear logic of presentation of theoretical aspects and examples of them, so the material will be understandable even to people far from mathematical calculations. This work should be read, first of all, by those whose work is related to communication: negotiations, large sales, consulting on various types of services, etc.
The value of this tutorial is the presence of a small workshop on the last pages - there are several tasks aimed at training thinking. The publication concludes with an impressive list of references, which will be useful for anyone wishing to continue their study of game theory.
Read the work “Game Theory. The Art of Strategic Thinking” is pleasant not only from the point of view of educational material, but also for the purpose of entertainment - the game has even penetrated into the style of writing. The authors encourage the reader to think and search for the right options among all possible ones, and this is a necessary condition for a successful player.
On our literary website books2you.ru you can download the book “Game Theory” by Barry J. Nalebuff, Avinash Dixit. The Art of Strategic Thinking in Business and Life" for free in formats suitable for different devices - epub, fb2, txt, rtf. Do you like to read books and always keep up with new releases? We have large selection books of various genres: classics, modern fiction, literature on psychology and children's publications. In addition, we offer interesting and educational articles for aspiring writers and all those who want to learn how to write beautifully. Each of our visitors will be able to find something useful and exciting for themselves.
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Description of the book: This book will definitely appeal to readers who like mathematical science. It will be useful both for those who are simply interested in it as a hobby, and for those who are planning to enroll in mathematics departments. The book is entertaining and captivating. It presents numerous theories as well as practical exercises. Readers will be able to learn something new every time and apply it in practice by solving a variety of mathematical exercises. With a book like this you will never be bored again. Numbers, formulas, theoretical knowledge - all this is collected here.
In the current times of active fight against piracy, most of the books in our library have only short fragments for review, including the book Strategy games. Available tutorial in game theory. Thanks to this, you can understand whether you like this book and whether you should purchase it in the future. Thus, you support the work of writer Avinash Dixit by legally purchasing the book if you liked it summary.
Strategic games - a classic textbook on game theory: clear definitions, questions, exercises, glossary, accessible presentation. Strategic thinking is the ability to analyze interactions with other people while they are analyzing the same situation in the same way. Game theory is the science of such interactive decision making. In other words, it is the science of rational behavior in interactive situations (that is, in the presence of other players). After studying the book, you will understand the general principles of analyzing strategic interactions, which will help you make informed and rational decisions in business and life.
Avinash Dixit, David Reilly and Susan Skeete. Strategy games. – M.: Mann, Ivanov and Ferber, 2017. – 880 p.
Download the abstract (summary) in or format (the abstract is about 4% of the volume of the book)
Part I. General principles
Chapter 1: Basic Concepts and Examples
Strategic thinking is the ability to analyze interactions with other people while they, in turn, do the same. You must take your opponent's plans into account, just as he takes yours into account. Game theory is the analysis or, if you prefer, the science of such an interactive decision-making process. When you choose your actions to achieve maximum success based on your own criteria, you are considered to be acting rationally. Game theory adds another dimension to the concept of rational behavior, namely interaction with other, equally rational decision makers. In other words, game theory is the science of rational behavior in interactive situations.
Good strategists use game theory in combination with their experience; we can say that playing strategic games is no less an art than a science.
Chapter 2. Approach to the analysis of strategic games
We make a distinction by using the term strategy games interaction between mutually aware players and the term solutions situations in which each person is free to make choices without worrying about the reactions or responses of others. In reality, game theory has a much broader scope. Many situations that begin as an impersonal market with thousands of participants turn into strategic interactions between two or more participants. This happens for one of two reasons: mutual obligations or personal information.
Classification of games based on answers to the following questions:
- Are the moves in the game performed sequentially or simultaneously?
- Does one player's gain mean another's loss? If so, the situation is called a zero-sum game. Trade or economic activity offers ample opportunities for transactions that benefit everyone.
- Is the game played once or repeatedly, with the same or changing opponents?
- Is there inherent uncertainty in the game regarding external conditions and/or the opponent's strategy? Complex strategic situations arise when one player knows more than the other, and are called games with asymmetric information.
- Are the rules of the game fixed or can they be manipulated?
- Is it possible to enforce cooperation agreements? If yes, then the games are called cooperative.
Terms.Strategies- These are the choices available to players. The strategy must include a comprehensive action plan. Strategy is a plan for executing a sequence of actions in response to changing circumstances resulting from actions taken by other goal-oriented players.
Winning The number corresponding to each possible outcome of the game is called. If a player is faced with a random set of game outcomes, then the expected payoff is said to be equal to the average of the wins for each individual outcome, weighted by their probability. If we include the attitude of players to risk into the conceptual model of game theory, we can talk about the expected utility method.
Rationality. In most cases, game theory is based on the assumption of rational behavior, which has two components: an understanding of one's own interests and the calculation of actions that best correspond to them. Rationality does not mean that players are selfish, or that players think in the short term. Being rational does not mean having the same value system as other players. Typically, players are not even aware of other players' value systems; this is one of the reasons why many games are classified as games with incomplete or asymmetric information.
General knowledge of the rules. We believe that at some level players have a common understanding of the rules of the game, which consist of: 1) a list of players; 2) the strategy available to each player; 3) the winnings of each player for all possible combinations of strategies that all players adhere to; 4) the assumption that each player is a rational maximizer.
Concept equilibrium implies that each player uses a strategy that is the best response to the strategies of the other players. We formulate game-theoretic concepts of equilibrium in Chapters 37 and then use them in subsequent chapters.
Observation and experiment. Any theory must relate to reality in two ways. Reality should help structure the theory and provide verification of its results. Two methods allow us to determine the real characteristics of strategic interactions: 1) observing them in natural conditions and 2) conducting special experiments.
We bring to your attention three functions of game theory. Explanation. Many events and their consequences make us ask the question: why did this happen? Forecasting. Consultations or recommendations.
Part II. Concepts and methods
Chapter 3. Games with sequential moves
Players in sequential-move games need to consider the consequences of their current moves before choosing actions. Typically, analysis pure games with sequential moves requires building a game tree. Such a tree consists of nodes and branches that display all the probable actions of each player at each opportunity to make a move, as well as payoffs for all expected outcomes of the game (Fig. 1). Each player's strategy is a comprehensive plan that describes his actions at each decision node depending on all possible combinations of actions taken by other players at previous nodes. The end nodes list the payoffs of our four heroes in this order: Ann, Bob, Chris, Deb.
Rice. 1. Game tree example
Pay your attention to the “Nature” node. In it, a random event occurs, such as a coin toss, and the outcome of the game will depend on whether it lands on heads or tails. Using a player called “Nature” allows us to introduce a factor of external uncertainty into the game and puts at our disposal a mechanism that makes it possible for events to occur that are beyond the control of the actual participants in the game.
When thinking about a game tree, you should always start by looking at the action nodes that lead directly to the leaf nodes. Using late-game choices to predict the consequences of earlier actions allows choices to be calculated at nodes preceding the final decision nodes. Then the same can be done with the previous nodes, etc. By moving backwards through the game tree in this way, you can solve the entire game. This inverse reasoning method.
When all participants in the game use the method of inverse reasoning to select optimal strategies, such a set of strategies is called reverse equilibrium. Most games have exactly one such equilibrium.
Most people who have no experience in playing strategic games are of the opinion that first mover advantage must be present in all games. However, this is not true. First mover advantage depends on a player's ability to commit to an advantageous position and force other players to accommodate him; second move advantage due to the flexibility of the player making the second move to adapt to the choices of other players. When there is a first or second move advantage in a game, each player can try to manipulate the order of play to secure an advantageous position.
Centipede game. The experimenter places a 10-cent coin on the table. Player A can take it or skip his turn. If player A takes the coin, the game is over; in this case, A receives 10 cents, and B receives nothing. If player A misses his turn, the experimenter places another 10-cent coin on the table, and now player B has to choose whether to take both coins or miss his turn. Players take turns, and the stack of coins grows until it reaches one dollar (Fig. 2).
Rice. 2. Game tree Centipede
Let's analyze the game using reverse reasoning. Obviously, Player B will take one dollar in the last step, so Player A should take 90 cents in the penultimate step, and so on. Therefore, player A must take the 10 cent coin at the very beginning and end the game. However, during experiments, such games usually last several rounds. Through irrational behavior, players as a group gain more money than if they had followed the reverse logic. In a classroom experiment conducted by one of us (Dixit), one such game reached the very end. Player B took his dollar and completely voluntarily gave 50 cents to player A. Dixit asked: “Are you in agreement? Are you and B friends?” To which player A replied: “No, we didn’t even know each other before. But now he is my friend."
The game points to possible problem with the logic of inverse reasoning in non-zero-sum games. Note that by skipping the first round, Player A is already showing that he is not relying on backward reasoning. So what should Player B expect from him in the third round? Having missed a turn once, Player A may do so again, which means that it would be advisable for Player B to miss a turn in the second round. A player may deliberately skip a turn in one of the initial rounds of the game to signal a willingness to skip turns in future rounds. This problem does not arise in zero-sum games, in which there is no incentive to cooperate through waiting.
Chapter 4. Games with simultaneous moves: discrete strategies
It is convenient to describe games with simultaneous moves and discrete strategies using a game table (Fig. 3). In a two-player game, the table has two dimensions, and the row and column headings are the strategies available to the first and second players. The cells indicate the winnings that players will receive with the appropriate configuration of strategies. Games with three players require a three-dimensional table. It is generally accepted that of the two numbers representing winnings, the first number corresponds to the winning of the Row, and the second - to the winning of the Column.
Rice. 3. Representation of a game with simultaneous moves in the form of a table
Nash equilibrium in Fig. 3 corresponds to the Low row and the Middle column. A row cannot increase its winnings by moving up or down the table. Likewise, a Column cannot increase its winnings by moving left or right. The Nash equilibrium in a game is a list of strategies (one for each participant) in which no player can increase his payoff by choosing another strategy from those available to him if other players adhere to the strategies specified in this list.
The Nash equilibrium is not always optimal for both players. In Fig. 3 pair of strategies Bottom, Right provides payoffs 9, 7. However, playing independently of each other, players will not be able to adhere to these particular strategies. Getting wins of 9, 7 will require cooperative action.
If each player's best choice does not depend on the correctness of his beliefs about the other player, we speak of dominant strategies for both players.
Some player strategies may be dominated, even if no one strategy dominates the others. Sequential, or iterative, elimination of dominated strategies amounts to removing them and reducing the size of the game until further reduction is no longer possible. For example, in Fig. 3 Row has a dominated strategy High, which is dominated by a strategy Below (across all columns). This allows you to delete the High line.
Likewise, the Left Column strategy is dominated by the Right Column strategy. Note that we could not draw this conclusion earlier, before removing the Row High strategy. In the context of the remaining set of strategies (Row Top, Low, and Bottom and Column Middle and Right), the Row Top and Bottom strategies are dominated by the Low strategy. When Row's only strategy is Low, Column will choose its best response Middle. Therefore, this game is dominance solvable and its outcome is Low/Middle with payoffs of 5, 4. We define it as a Nash equilibrium.
Other games may not be dominance solvable, and consistently eliminating dominated strategies may not provide a unique outcome to the game.
If eliminating dominated strategies does not allow one to find a Nash equilibrium, one can apply best answer analysis. What's the best answer of this player for every choice the other player can make? In Fig. 4 we have circled the best answers. In the “low” / “middle” cell there are two selections. Therefore, the strategies “low” for Row and “in the middle” for Column will be the best responses to each other. We have found the Nash equilibrium in this game once again.
Rice. 4. Analysis of the best answers
When best response analysis in a discrete strategy game does not reveal a Nash equilibrium, it means that the game does not have a pure strategy equilibrium.
Games do not necessarily have a single Nash equilibrium. There is a class of games that can be denoted as coordination games. Their participants have common interests, but because the players act independently of each other, coordinating the actions needed to achieve a common preferred outcome is problematic. Successful coordination requires that the desired outcome be the focal point. The players' expectations should converge at this point: everyone should know that everyone knows that... everyone is making this choice. This is the function that many public institutions and agreements perform.
In some games there is no Nash equilibrium in pure strategies.
Chapter 5: Simultaneous Move Games: Continuous Strategies, Analysis and Discussion
Consider the example of price competition. The owners of two restaurants must set food prices to maximize profits. Since prices can take any value within an (almost) infinite range, we find the rules for best answers and use them to solve the game and determine equilibrium prices. Let's denote the price of restaurant 1 as P x, and 2 as P y. Let’s say that servicing one client costs each restaurateur $8. Assume that experience or market research shows that the number of customers, Q x and Q y (in hundreds of customers per month) is given by the equations
Q x = 44 – 2P x + P y
Q y = 44 – 2P y + P x
The basic idea behind these equations is that if one restaurant increases its price by $1 (say P y rises by one dollar), its sales will decrease by 200 per month (Q y will decrease by 2), and the other restaurant's sales will will increase by 100 per month (Q x will increase by 1). We can assume that 100 customers will move to another restaurant, and another 100 will stay at home.
Profit is calculated as the product of net income per client (price minus service costs, or P x - 8) and the number of clients served:
π x = (P x – 8)Q x = (P x – 8)(44 – 2P x + P y) = –8(44 + Р y) + (16 + 44 + Р y)Р x – 2( Р x) 2 = –8(44 + Р y) + (60 + Р y)Р x – 2(Р x) 2
If we take the first derivative, we will find the value of P x at which π x reaches its maximum. π x → max at 60 + Р y – 4Р x = 0. Or Р x = 15 + 0.25Р y. Similarly, P y = 15 + 0.25 P x. In Fig. Figure 5 shows graphs of these two best answer rules.
Rice. 5. Graphs of best answers and equilibria in the game “restaurant pricing”
The point where the two best answer graphs intersect is the Nash equilibrium of the pricing game between the two restaurants: P x = P y = 20. In the equilibrium, each restaurant will charge $20 for the items on its menu and make $12 in profit for every 2,400 customers ( 2400 = (44 – 2 x 20 +20) x 100) served per month, which will provide a total profit of $28,800 per month.
In an oligopoly (small number of sellers), companies can negotiate prices. In this case, P x = P y and
π x = π y = (P – 8) (44 – 2P + P) = (P – 8) (44 – P) = –352 + 52P – P 2
Profit reaches its maximum at P = 26 (point best choice in Fig. 5). In this case, π x = π y = $32,400 per month. In the language of economics, an agreement to raise prices to a level that is optimal for both parties is called a cartel. High prices hurt consumers, so government regulators typically try to prevent cartels from forming and force companies to compete with each other.
In addition to the dominated ones, you can exclude strategies that may not be the best answer. The strategies remaining after such an elimination procedure are called rationalizable, and the concept itself is called rationalization. In such cases, we have a stronger justification for the Nash equilibrium, relying solely on rationality, without assumptions about the correctness of expectations. Consider the game in Fig. 6.
Rice. 6. Streamlined Strategies
Can Row believe that Column will choose strategy C4? It must be based on the Column's beliefs regarding the Row's choice. Can they make strategy C4 Column's best answer? No. If Column believes that Row will play R1, his best response is C1. If Column believes Row will play R2, his best response is C2. If Column believes that Row would prefer R3, then its best answer is C3. And if Column is convinced that Row will choose R4, then his best answer is either C1 or C3.
Therefore, C4 may not be Column's best answer. This means that Row, knowing that Column is rational, will in no case attribute the choice of strategy C4 to him. Therefore, Row should not proceed from the belief that Column will play C4. Note that while strategy C4 may not be the best response, it is not dominated by strategies C1, C2, and C3. Thus, "a strategy that may not be the best response" is a more general concept than "a dominated strategy."
The Nash equilibrium in this game boils down to each player choosing the number 0. In reality, the game is solvable by dominance. Even if each participant indicated 100, half of the average cannot exceed 67, so for each player the choice of a number greater than 67 is dominated by the choice of 67. However, this should be clear to all rational players, which means that the average cannot exceed 67, and two-thirds of that is 44, so any choice of a number greater than 44 will be dominated by a choice of 44. This process of iteratively removing dominated strategies continues until only the number 0 remains.
However, when a group plays this game for the first time, the winner is not the one who chooses the number 0. Typically, the winning number falls between 15 and 20. The most common numbers players choose are 33 and 22, which suggests that many of them perform only one or two cycles of iterative dominance, without continuing this process further. In other words, “level 1” players believe that everyone else in the game will choose numbers randomly, with an average value of 50, so the best answer is two-thirds of this number, that is, 33. Similarly, “level 2” players assume that all other players are reasoning at "level 1", so they choose two-thirds of 33, or 22, as the best answer.
One of the first areas of application of the concept of Nash equilibrium in relation to the behavior of subjects real world became the sphere of international relations. Thomas Schelling was the first to use game theory to explain phenomena such as the escalation of the arms race (see). Game-theoretic models based on the concept of Nash equilibrium provide a better understanding of the main factors of competition compared to older models based on perfect competition and estimated supply and demand curves.
Imagine that you are a farmer and your work depends on the whims of the weather. If the weather is favorable for a good harvest, you will receive an income of $160,000. If adverse weather conditions occur, your income will be only $40,000. You could try to reduce the risk by asking someone else to take on some of the risk. Of course, in exchange you will have to give something to this person. This like-kind exchange usually takes two forms: a cash payment or a reciprocal exchange or risk sharing.
The idea that there is a price for risk and a market for risk underlies almost all financial mechanisms in modern economies. For example, derivatives are just a way to distribute risk among those who are willing to bear it for the minimum price (see). Financial markets stimulate entrepreneurship by facilitating risk trading.
Asymmetric information. Manipulating information about your abilities and preferences known to other players allows you to influence the equilibrium outcome of the game. As a result, this manipulation of asymmetric information becomes a strategic game in itself. A more informed player can take the following actions: hide or give false information, reveal part of the true information. A less informed player can: obtain the necessary information or separate the truth from the lies; remain ignorant (ignorance of your opponent’s strategic move can protect you from his obligations and threats).
You know that others will form an opinion of you based on your actions, and because of this, you will try to come up with and take steps that will make them think that your information is trustworthy. Such actions are called signals, and the strategy for using them is called signaling.
If other players know more than you or perform actions that cannot be directly observed, you can use strategies that reduce this information gap. A strategy that forces the other player to reveal his information is called screening.
In many games, one of the participants knows something about the outcome of the game that others do not. For example, a seller of a used car knows a lot about it thanks to long-term operation, and a potential buyer can best case scenario obtain a minimum of information during a car inspection. In such situations, direct communication does not ensure reliable transmission of information.
If an insurance company offers a policy that costs 5 cents for every dollar of coverage, it will be especially attractive to people who know that their own risk (of illness or car accident) is greater than 5%. Of course, some people who know that their risk is below 5% will still buy such an insurance policy due to risk aversion. However, in the total population of persons applying for this insurance policy, the proportion of persons with more high degree risk will exceed the proportion of persons with a similar risk in the total population. Thus, the insurance company selectively attracts a disadvantaged, or unfavorable, group of customers. This phenomenon is known as adverse selection and is typical for transactions with asymmetric information.
The potential consequences of adverse selection for market transactions were demonstrated very clearly by George Akerlof in the paper that pioneered the economic analysis of asymmetric information situations and earned him the Nobel Prize in 2001 (see ). Signaling and screening strategies can overcome information asymmetry.
An insurance company may offer two insurance policies. The first provides a lower insurance premium, but provides coverage for a smaller percentage of the losses incurred by the client. The second policy provides a higher insurance premium, but also provides more high percentage insurance coverage for losses. Customers in a higher risk category choose policies with high premiums and high coverage, while customers in a lower risk category choose policies with lower premiums and low coverage.
A company that knows that its product is of high quality can send a reliable signal to potential buyers about this - give a guarantee. For example, Hyundai in the US market offered a 10-year, 100,000-mile warranty on its vehicles in the mid-1990s.
Companies can successfully set different prices for different groups consumers using screening tools. Such strategies are known in the economic literature as price discrimination. For example, airlines set different prices for refundable and non-refundable tickets and let travelers choose their own fare type. This pricing strategy is an example screening through self-selection.
When simply asking questions is not enough to obtain truthful information, a diagram may be needed screening. Screening achieves the desired results only when the screening tool encourages other players to reveal truthful information about their type; separation of types is possible only if there is compatibility of stimuli. Sometimes reliable signaling or screening may not be possible; in such a case, the equilibrium may entail a union of types or a complete collapse of the market or transaction for one of the types is likely.
In an equilibrium game with asymmetric information, players must not only use their best actions given the available information, but also make correct inferences (update information) while observing the actions of other players. This type of equilibrium is known as Bayesian Nash equilibrium.
Chapter 9. Strategic moves
If the rules of the game are not fixed from the outside, each player has an incentive to manipulate them in order to ensure a more profitable outcome for themselves. Tools that allow you to manipulate the game in this way are called strategic moves.
Strategic move changes the rules original game in order to create a new two-stage game. The different actions performed in the first stage correspond to different strategic moves; We will divide them into three categories: obligations, threats and promises. The goal of all three is to change the outcome of the second stage of the game in their favor.
Commitment is simply taking advantage of first mover advantage if one exists. Of course, for this to happen, the commitment must be credible. In order for your strategic move to be effective, you must do something early in the game to ensure credibility - something that will show your opponent that under no circumstances will you deviate from the agreed action.
Note that threats and promises are response rules: your future actual action depends on what the other players do, but your freedom of action is further limited by mandatory compliance with the stated rule. The goal is to change the expectations (and therefore the actions) of other players to your advantage. Threat is a response rule that results in negative consequences for other players if they act against your interests. Promise- a response rule, according to which you offer to provide other players with a positive outcome if their actions are consistent with your interests.
Example of a threat: trade relations between the United States and Japan. Each country can keep its markets either open or closed to the other country's goods. But the preferences of the two countries regarding the outcome of this game are somewhat different (Fig. 10).
Rice. 10. Table of winnings in the trading game between the USA and Japan
The equilibrium outcome is “open American market” / “closed Japanese market”, and the payoffs are 3, 4. But let’s say the US chooses the following conditional rule answer: “We will close our market if you close yours.” As a result, we get a two-stage game (Fig. 11). It will lead to Japan opening up the market, and the US will get the best outcome.
Rice. 11. Tree of a trade game between the USA and Japan using a threat; Nash equilibrium highlighted
Carrying out a threat in the true strategic sense must necessarily be costly to the one who makes it, and the action that constitutes the essence of the threat must cause mutual harm.
We identify two approaches to ensuring the credibility of strategic moves: 1) limit your own freedom of action in the future so that you have no choice but to carry out the actions prescribed by your strategic move; 2) change your own future payoffs in such a way that performing the actions prescribed by the strategic move is optimal for you. For example, in trade policy, automatic procedures for imposing retaliatory tariffs on imports when another country attempts to subsidize its exports to your country are common.
You can create for yourself reputation a person (company, country) who always fulfills threats and promises. Reputation comes from the fact that when you're away from home, you choose to eat at a restaurant chain you know rather than risk going to a local restaurant you don't know. In practice, credibility is not an all-or-nothing situation, but a matter of degree.
Salami tactics is a tool that allows you to reduce the size of your opponent's threat the same way you cut salami: one slice at a time. You don't comply with another player's wishes that much small degree(whether in the case of deterrence or coercion) that it makes no sense for him to take any radical action in response. If your move is effective, you commit another small violation, then another, and so on.
Chapter 10. Prisoners' Dilemma and Repeated Games
The couple is suspected of murder. They are interrogated separately, and each of them can either admit to committing a crime or completely deny their involvement in it (Fig. 12). Winnings are calculated in years of prison time; therefore, low values are more beneficial to both players.
Rice. 12. Payoff table for the standard Prisoner's Dilemma game
The prisoners' dilemma is a non-cooperative game; players make decisions and implement them separately from each other. At the same time, there are mechanisms to maintain cooperation. Most often the latter can be achieved through repeated play. Each player may fear that one instance of non-cooperation will lead to its cessation in the future. If the value of future cooperation is great enough to exceed the short-term benefit of not doing so, then players' long-term self-interest may automatically deter them from cheating without any need for additional punishment or pressure from third parties.
In repeated games, players can choose strategies based on behavior in previous rounds of the game. Such strategies are known as conditional strategies. Most of the latter fall into the category of trigger strategies, in which the player cooperates as long as the opponent also does so, but any deception on the part of the latter “triggers” punishment. For example, in a tit-for-tat strategy, the player chooses to cooperate in the first round of the game, and then in each subsequent round chooses the actions chosen by the opponent in the previous round.
Trigger strategies are determined by the number of game rounds: whether it is finite or infinite, and whether this number is known in advance. For example, in bad times, when an entire industry is on the verge of collapse and companies feel that they have no future, competition may become significantly more intense (cooperative behavior may be less common). On the other hand, when fashion changes for products produced by an unchanged group of companies maintaining long-term relationships, the partnership remains.
In addition to repetition, there are other tools for solving the prisoners' dilemma. Players can be directly penalized if they refuse to cooperate. In this case, the option of “turning in an accomplice” loses its attractiveness. Another method of solving the prisoners' dilemma refers to situations in which one player takes on the role of leader in the interaction. In real strategic situations, one player may be relatively “large” (the leader). For example, Saudi Arabia played a stabilizing role in OPEC for many years: to maintain high oil prices, it reduced its production, while one of the smaller producers (such as Libya) increased it.
In laboratory experiments, it was found that the equal response strategy, which has the properties of predictability, benevolence, retribution, and forgiveness, on average produces very good results in a repeated prisoners' dilemma.
Chapter 11. Group games
Multi-player games have problems collective action. There are three types: the prisoners' dilemma, the coward game and the trust game. Winnings in such games are classified as non-excludable benefits: A person who has not contributed to its implementation cannot be prevented from benefiting from it. Often games with many participants would be more correctly called games with collective inaction.
The common characteristic of all these games is that their participants must decide whether to use a particular shared resource, be it a highway, a high-yield investment fund, or a pond with a lot of fish. Such group games with "inaction" are better known as games with the allocation of common resources: the total payoff of all participants is maximized when they refrain from excessive use of common resources. The problem associated with the failure to reach the social optimum in such games is known as tragedy of the commons.
Let's describe the impact of each person's decisions on other people and the group as a whole. 8,000 suburban residents commute into the city every day to work. You can choose to travel on either a motorway (action P) or a local road network (action S). A trip on local roads invariably takes 45 minutes, no matter how many cars are traveling on them. A trip on the expressway takes only 15 minutes, provided there is no congestion. However, each driver who chooses the expressway increases the travel time of every other driver who takes that route by 0.005 minutes.
Winnings in the game are calculated in minutes of time saved - for example, how many minutes is the travel time less than one hour. Therefore, the payoff for drivers, denoted as S(n), who choose a route along local roads, is a constant value: 60 – 45 = 15, regardless of the value of n. The payoff for drivers who choose the expressway is P(n) = 45 – 0.005n (Fig. 13).
Rice. 13. Route selection game
Let's say there are 4000 cars on the highway. With so many cars on the road, each driver needs 15 + 4000 x 0.005 = 15 + 20 = 35 minutes to get to work; with each receiving a payoff of P(n) = 25. You may decide to switch from driving on local roads to traveling on the expressway. Choosing a new route will increase the value of n by 1. Now the number of drivers who chose the highway is 4001 (including you), and the travel time for each is 35 + 5 / 200, or 35.005 minutes. In this case, each driver will receive a payoff P(n + 1) = P(4001) = 24.995, which still exceeds the payoff from traveling on local roads. Therefore, you have a personal incentive to change the route because P(n + 1) > S(n) (24.995 > 15).
Choosing a different route brings you a personal benefit (that only you receive) equivalent to the difference between your winnings before and after such a switch; it is P(n + 1) – S(n) = 9.995 minutes. We call her marginal (additional) personal benefit. However, now, because of your decision to change the route, each of the 4,000 other drivers who took the highway will have to spend 0.005 minutes more on their trip. The total impact of your decision on all other drivers is 4000 x (0.005) = 20. Your action, that is, moving from local roads to the expressway, influenced the winnings of other players. Whenever one person's action has a similar effect on other people, there is a spillover effect, or spillover, or externality.
We call the combination of marginal personal benefit and externality marginal social benefit. The latter in our example is 9.995 – 20 = –10.005 minutes. Therefore, the overall social effect of your switching route is negative. However, a person changing his commute route does not take into account the spillover effect (externality); he is motivated only by his own winnings.
How to ensure optimal distribution of drivers from the point of view of society as a whole? Different cultures and political groups use different systems, each with their own advantages and disadvantages. Society could simply ban 3,000 drivers from the expressway. But by what criteria are they selected? A bureaucratic society could set criteria based on officials' complex calculations of need and merit, and each driver would then take costly steps to meet those criteria. A politicized society may favor important "independent voters", or organized groups activists, or people making donations. In a corrupt society, those who bribe officials or politicians could gain privileges.
There is a scheme whereby you are only allowed to drive on the motorway on certain days, depending on the last digit on your car's license plate. However, this arrangement is not as democratic as it may seem at first, since rich people can buy two cars and choose license plates so that they can use the highway on a daily basis.
Many economists prefer to introduce tolls. This clearly demonstrates to each driver the additional costs that his actions entail, which in turn encourages him to choose the socially optimal action. Economists in this case say that an individual is forced adopt externality.
There are also positive side effects. For example, vaccination. Each person who gets vaccinated reduces both his own risk of contracting the disease (marginal personal benefit) and the risk of others contracting it from him (spillover effect).
In games played in large groups, there is a diffusion of responsibility, which can determine behavior when an individual waits for others to perform the necessary action, and he takes on the role of a “free rider”, that is, he benefits from this action. When someone needs help, the likelihood of it being provided decreases as the size of the group of people who can provide it increases.
Chapter 12. Evolutionary games
So far we have assumed that each player makes a conscious and thoughtful choice from the strategies available to him. However, recent theories have cast doubt on this assumption. The most valid and convincing criticism comes from psychologist and 2002 Nobel Prize winner in economics Daniel Kahneman (see). In his opinion, people have two different decision-making systems. System 1 is instinctive and fast, system 2 is calculating and slow.
This implies a completely different way of conducting and analyzing games. Players enter the game with Instinctive System 1 and play the strategy it tells them, even though that strategy may not be optimal. A positive result reinforces the instinct, while a negative result contributes to its gradual change. Where does this process of interactive dynamics of instincts lead?
The biological theory of evolution is based on three fundamental principles: heterogeneity (heterogeneity), fitness and selection. Animal behavior is genetically determined: a complex of one or more genes ( genotype) determines the pattern of behavior (behavioral phenotype). The natural diversity of the gene pool ensures the heterogeneity of phenotypes in the population. Some behavior patterns are more consistent with prevailing conditions than others; the success of a phenotype is expressed as a quantitative indicator called fitness.
Reproductive success allows an animal to pass on its genes to the next generation and maintain its phenotype. The more fit phenotypes then become relatively more abundant in the next generation than the less fit ones. It is this dynamic selection process that changes the combination of genotypes and phenotypes.
From time to time, new genetic mutations arise spontaneously. Many of them create behavior patterns (phenotypes) that do not fit well with their environment and therefore die out. However, sometimes a mutation leads to the formation of a new phenotype that is more adapted to the environment. Such a mutant gene can take over the population, that is, form a significant proportion of it. Biologists call the configuration of a population and its current phenotypes evolutionarily stable, if no mutant phenotype can successfully capture it.
In interactions between people, strategy can be embedded in a person's mind for various reasons, including not only genetics, but also socialization, cultural upbringing, education, or empirical experience based on past events. All this can be covered by Kahneman's instinctive, fast system 1. A population may consist of a collection of different people with different backgrounds or experiences that lead them to pursue different System 1 strategies.
The gradual process of change, taking into account outcomes, experience, observations and experiments, forms the dynamics of a calculated, slow system. 2 There are two types of evolutionarily stable configurations of biological games. First, one phenotype may be more fit than others, and the population may consist only of it. Such an evolutionarily stable result is denoted by the term monomorphism. In this case, one dominant strategy is called evolutionarily stable strategy.
Second, two or more phenotypes may have the same level of fitness, so they may coexist in certain proportions. Then they say that the population exhibits polymorphism. Polymorphism is very close to the game theory concept of mixed strategy. However, there is one important difference. In polymorphism, different players pursue different pure strategies, but the population as a whole exhibits a mixture of strategies. If there is a strictly dominant strategy in a game, it will necessarily be evolutionarily stable.
An evolutionarily stable strategy must be a Nash equilibrium in a game played by consciously rational players with the same payoff structure. Thus, the evolutionary approach provides an indirect justification for the rational approach.
Chapter 13. Development of mechanisms for the principal-agent problem
Usually the less informed player is called principal, and the more informed - agent. The process used by the principal to create the right set of incentives for the agent is known as development of mechanisms.
Many restaurants offer three-course prix fixe menus and inexpensive set meals along with regular a la carte dishes. This strategy allows the restaurant to distinguish different types of customers who prefer different soups, salads, entrees, desserts, etc. Book publishers typically sell new hardcover books first and publish the paperback version only a year later. Often the difference in price between the two versions is much greater than the difference between the cost of the two types of books. This pricing scheme is designed for two types of buyers: those who want to read a book as quickly as possible and are willing to pay more for it, and those who are willing to wait for a better price. There are many examples of such screening mechanisms of price discrimination.
The second type of mechanism design problem involves moral hazard. Suppose you are the owner of a company starting a new project and need to hire a manager to oversee its implementation. If you don't have a way to track the manager's efforts, you need to incentivize him to see the project through successfully, for example by paying a bonus upon completion.
Insurance markets are also subject to moral hazard. Insurance companies need to figure out how to write acceptable insurance contracts that incentivize customers to take actions that reduce the likelihood of them filing an insurance claim. For example, insurance companies would like the people they sell health insurance policies to to get regular preventive health checkups, and the people they sell auto insurance policies to to continue to practice safe driving. Most insurance policies leave a portion of the policyholder's risk uninsured in order to reduce moral hazard.
Can a manager's optimal incentive system always be determined by base salary and profit sharing? No. Given three possible outcomes (project failure, moderate success, and great success), the percentage premium for going from failure to moderate success may not be the same as the premium for going from moderate to great success. Therefore, the optimal incentive system may be nonlinear. But this incentive system is not without its drawbacks.
For example, mutual fund managers are often rewarded for superior performance over a calendar year. It is paid at the expense of the company in the form of premiums, as well as at the expense of investors who invest money in the corresponding fund. If these compensation structures are nonlinear, managers will increase the risk level of their fund's investment portfolio.
When one worker's earnings depend on the profits of the entire company, each individual employee sees only a weak connection between his efforts and the overall profits, with each individual receiving only a small share of it. And this share is a very weak incentive to make increased efforts to fulfill their duties. Even in small teams, each member may be tempted to shirk their work and take advantage of the fruits of their colleagues' labors.
The outcome of each agent's task depends partly on his efforts and partly on chance. This is why performance-based incentive schemes often put the agent's payoff at risk.
Another way to ensure worker motivation is for the company to pay the worker a wage above the generally accepted rate, and the difference between the two rates represents the worker's surplus, or economic rent. The employee receives it on the condition that he performs his duties conscientiously, but if he begins to act out, this may be discovered and he will be fired. As a result, he will have to return to the general labor market, where he can only receive the generally accepted wage.
Part IV. Application of game theory in specific strategic situations
Chapter 14. Balancing on the Brink: The Cuban Missile Crisis
Balancing on the brink is a type of strategic move. You need to take an action in advance that creates the likelihood (but not the inevitability) that if your opponent ignores your threat, there will be consequences that will be detrimental to both parties. A formal description of the Cuban missile crisis can be obtained by constructing a game tree (Fig. 14).
Rice. 14. A model for overcoming the Cuban missile crisis using a simple threat
We can find perfect balance. Faced with a US threat, the USSR would receive a payoff of –4 if it withdraws its missiles and –8 if it refuses to do so, so the USSR will choose the former. Having analyzed this outcome in advance, the United States expects to receive a payoff of 1 if the threat is issued, and -2 if it is not; therefore, it is more profitable for the United States to put forward a threat, since this outcome provides them with a payoff of 1, and Soviet Union–4. However, such an interpretation of the crisis is unsatisfactory: why then did the Soviet Union even need to place missiles in Cuba if it could have foreseen such a development of the game and understood how it would end?
Almost all games have an element of uncertainty. You cannot know for sure your opponent’s value system and cannot be completely sure that he will accurately perform the required actions. Therefore, the threat contains a double risk. Your opponent may ignore it, and you will have to perform the action that constitutes the essence of the threat; or he may comply, but the threat will still be carried out in error. When such risks exist, the consequences of the threat for the player making it become an important factor.
The Cuban missile crisis is rife with such uncertainties. Graham Allison reveals all these difficulties and uncertainties in his excellent book, The Essence of Decision. After analyzing them, Allison comes to the conclusion that the Cuban Missile Crisis cannot be explained in terms of game theory, and proposes two alternative options interpretations: one is based on the fact that the bureaucracy has its own established rules and procedures, and the other is built on the internal politics of the United States and the Soviet state and military apparatus. According to Allison, the political explanation is the most acceptable.
Brinkmanship is a strategy whereby you expose your opponent and yourself to a progressively increasing risk of mutual harm. The actual occurrence of a harmful outcome is not entirely under the control of the person making the threat. In most confrontations (for example, between a company and a union, a husband and wife, a parent and a child, the President and Congress, etc.), one side cannot be sure of the goals and capabilities of the other. Therefore, most threats involve a risk of error, and every threat must contain an element of brinksmanship.
Chapter 15. Strategies and Voting
Vote aggregation methods can be categorized by the number of options, or candidates, considered by voters at any given time. Binary methods involve choosing one of two alternatives at a time. During an election with exactly two candidates, the candidate who receives the most votes wins. If there are more than two alternatives, paired voting can be used - voting on pairs of alternatives during several rounds according to the principle of relative majority to determine the most preferable alternative (see).
Multiple methods allow voters to consider three or more alternatives simultaneously. One group of multiple voting methods involves using information about the position of alternatives on the ballot to determine the number of points taken into account in calculating the voting results; such voting methods are known as positional methods. The principle of relative majority voting is a special case of the positional method, when each voting participant casts one vote for his most preferred alternative. When counting votes, she is awarded one point; the winner is the alternative that receives greatest number votes (points).
Condorcet's Paradox is one of the most famous and important voting paradoxes. As noted earlier, according to the Condorcet method, the winner is the candidate who receives the majority of votes in each round of paired comparisons. Condorcet's Paradox occurs when this process fails to determine a winner.
Analysis of voting paradoxes suggests that voting methods have a number of shortcomings. Is there a voting system that satisfies certain regularity conditions, including the transitivity condition, that is the most “fair”, that is, most accurately takes into account the preferences of the electorate? Kenneth Arrow's Impossibility Theorem tells us that the answer to this question is no.
Many voters consider supermajority elections to be the fairest, but such elections nevertheless open up many opportunities for strategic behavior. For example, in presidential elections there are usually only two viable candidates to win, and when there is a relatively small gap between them, a third candidate may enter the race to deprive the leading candidate of the vote; if the third candidate actually reduces the leader's chances of winning, he is called spoiler.
In politics, a spoiler is a candidate or party in an election who has no chance of winning, but who diverts some of the votes for another candidate with a similar program, thereby increasing the chances of victory for a candidate or party with an opposing position on major issues. Ross Perot played such a role during the 1992 US presidential election.
A strategic analysis of the behavior of two candidates participating in the elections states that both candidates will position themselves in the political spectrum in the same place as median voter. Three characteristics of the equilibrium in the candidate positioning game can be noted. First, they are both positioned in equilibrium in the same position. This illustrates principle of minimal differentiation is the overall outcome of all two-player games that involve competition for location, whether presidential candidates select a political platform, or street vendors select the location of a hot dog cart, or electronic device manufacturers select product features.
Secondly, both candidates are in the position of the median voter. Third, the position of the median voter does not always coincide with the geometric center of the political spectrum. These two positions coincide if the distribution of voters is symmetrical, but the median voter may be located to the left of geometric center, if the distribution is skewed to the left, and to the right, if the distribution is skewed to the right.
Another paradoxical result is that the outcome of any given election, given a set of voter preferences, can change depending on the voting procedure used.
Voters may use strategic behavior in the game that provides voting choice or in the election itself by distorting their preferences. Voters may strategically distort their preferences to achieve the most desirable outcome or to avoid an undesirable outcome. Given imperfect information, voters may decide whether to vote strategically based on their beliefs about the behavior of other voters and their knowledge of the distribution of their preferences.
Chapter 16. Bidding strategy and auction structure
The term “auction” refers to any operation in which the final price of an item offered for sale is determined through competitive bidding. Auctions are characterized by the presence of information asymmetry between the seller and the buyer, as well as between the buyers participating in the auction. In this regard, signaling and screening can become important elements of the strategy of both buyers and sellers.
Auctions vary in how bids are submitted and how the final price the winner pays is determined. These aspects of the auction, which are set in advance by the seller, are called auction rules. In addition, auctions can be classified by the type of object offered for sale, as well as by the method of its valuation; this defines the auction environment.
In most cases, the auction rules are determined by the seller, and he must do so with limited information about the buyer's willingness to pay. Thus, when choosing auction rules, the seller designs the auction mechanism (see Chapter 13). Auctions can be divided into open and closed.
Open auctions include the up auction, or English auction, and the down auction, or Dutch auction. In a first-price closed auction, the item for sale goes to the highest bidder and pays the bid price. In a second-price closed auction, the highest bidder receives the item for sale but pays the price specified in the second-highest bidder's bid.
The second auction is often called the “Vickrey auction”, named after the Nobel Prize winner in economics. Vickrey showed that under such rules, offering the true price is the dominant strategy of each bidder. In this regard, we jokingly call such an auction the Vickrey truth serum.
First-price closed auctions are similar to Dutch auctions, while second-price closed auctions resemble English ones.
The distinctive feature of the auction environment is based on the differences between objects with general and personal value. In the first case, the object put up for sale has the same value for all bidders, but each of them knows only its approximate value.
The winner's curse is a warning to bidders that if they win an auction and receive the item they are looking for, they have likely paid more for it than it is actually worth. This case is not much different from buying a used car (“lemon”). The theory of adverse selection in markets with asymmetric information is directly applicable to the described shared value auction.
The simplest experiment to test the winner's curse boils down to holding an auction for the sale of a jar of coins. The winnings in this game are objective, but each participant forms a subjective assessment regarding the number of coins in the bank, and therefore the size of the winnings (this is an example of an auction with a pure total value). Most teachers who conducted such experiments with students invariably found that the price offered was significantly inflated.
However, there is a price to pay for using the information retrieval mechanism. In a Vickrey auction, buyers reveal the truth about their valuations only because it brings them a certain profit. A closed second-price auction reduces the seller's profit.
If there is a neutral attitude to risk and independent assessments by buyers of the value of the object put up for sale, sellers can count on the same intermediate level income using any of the four main types of auctions: English, Dutch and closed first and second price auctions.
Chapter 17. Negotiations
All negotiation situations have two things in common. First, the total payoff that the negotiating parties can achieve as a result of reaching consensus must be greater than the individual payoffs that they could obtain separately, that is, the whole must exceed the sum of the parts. Second, negotiations are not a zero-sum game. If there is a surplus, they come down to dividing it.
Before the advent of game theory, theorists could not understand at a systemic level why one party in a negotiation gets more than the other, and attributed this to vague and unexplained differences in so-called bargaining power.
One game theory approach views negotiations as cooperative game, in which negotiators work together to find and implement a solution. Another approach views negotiation as a non-cooperative game in which negotiators choose strategies separately and seek an equilibrium.
Avinash Dixit, Susan Skeete and David Reilly Jr.
Strategy games. An accessible textbook on game theory
Scientific editor Alexander Minko
Published by permission of W.W. Norton & Company, Inc. and literary agency Andrew Nurnberg
All rights reserved. No part of this book may be reproduced in any form without the written permission of the copyright holders.
© W. W. Norton & Company, Inc., 2015, 2009, 2004, 1999
© Translation into Russian, publication in Russian, design. Mann, Ivanov and Ferber LLC, 2017
* * *In memory of my father, Kamalakar Ramachandra Dixit
Avinash Dixit
In memory of my father, James Edward Skeete
Susan Skeete
To my mother, Roni Reilly
David Reilly
Foreword by the publishing partner
Here is a logical continuation of the book “Game Theory” by Avinash Dixit and Barry Nalebuff. In the new book, we are again involved in the analysis of many situations, on the basis of which we learn to understand the types of different strategies and predict the behavior of game participants.
In Russia, game theory has become not just a fashionable topic, but also mandatory knowledge for people solving complex strategic problems. The demand for literature on this topic is growing, and here in front of you is one of the few worthy books in which the authors managed to clearly explain the concepts and techniques of strategic play.
An interesting point: strategy games are not limited to just mathematical analysis of the situation. One of the main conditions of the game is the presence of players, each of whom has their own desires, tasks and goals. And it is the combination of methods of psychology and mathematics that makes it possible to build the most successful game strategy.
Despite the fact that all people in one way or another encounter games in everyday life, there are those who need this knowledge in the first place. I'm talking about leaders. Their role is to achieve the goals of the department or organization, taking into account the many people involved in the process. Project management, negotiation, change implementation - each of these areas of business is a strategic game.
What does this look like in practice? Not long ago, Samolov Group consultants led a project to support negotiations for the head of a large construction company. Our client's task was to purchase a large plot of land for construction. It was difficult for him to negotiate with his partner on his own terms. Together with the customer, we analyzed the interests and capabilities of the opponent. After the collection, it was discovered that the partner was approaching the repayment period of a large loan. Our client proposed new terms of the transaction under which the opponent had the opportunity to repay this same loan. Thus, the contract was signed for the amount that our client expected. The example illustrates the importance of player analysis as it allows one to determine the most winning strategy.
The higher the level of a manager, the more important it is for him to develop strategic thinking and the ability to apply game theory methods in practice. In reality, Russian managers are accustomed to solving situations “here and now”, without thinking about long-term scenarios. Therefore, learning the skills of planning, setting goals, and developing strategies is quite difficult even for experienced managers. It is necessary to admit to yourself that if you do not develop these competencies and do not pay due attention to planning, it will be more difficult to achieve serious goals.
In the book “Strategic Games” the methodology is presented through cases. This format may be somewhat unusual for Russian readers. In Russia, training first involves a description of the methodology, and then its application. In the same book, cases are first given, and then an analysis and conclusion of the methodology is made from them. The text of the book may seem difficult to understand, but the information contained in it is of great value, so I recommend that the interested reader take the time to read and get intellectual pleasure from it.
Ivan Samolov,Commercial Director of Samolov GroupPreface
We wrote this textbook for teachers and first- and second-year college students to help them master the basics of game theory. It does not require prior knowledge in the fields in which this science is applied (such as economics, political science, evolutionary biology, etc.); School level mathematics is sufficient. We must say that the result obtained exceeded all our expectations. Today, many courses in this discipline are taught in places where they were unheard of 20 years ago, and some of them were developed under the influence of our textbook. And the emergence of competitors and imitators on the market is another convincing sign of success.
However, success is not a reason for complacency. In each subsequent edition of the textbook, we continued to improve the material presented in it, taking into account the comments and suggestions of teachers and students, as well as our own experience in using it.
The main innovations in the fourth edition are related to mixed strategies. In the third edition we treated this issue in two chapters based on the differences between simple and complex topics. Easy topics included solving and interpreting equilibria in mixed strategy games in 2 × 2 games, and the main difficult topic was general theory mixing in games with more than two pure strategies, when some of them may remain unused in equilibrium. However, we have found that few teachers address the second of these chapters. Therefore, we decided to combine simple topics and some basic concepts from more complex topics into one chapter on mixed strategies (Chapter 7). Some material not included in this chapter will be available to readers wishing to explore the topics in more depth. higher level complexity, in the form of online applications.
We've improved and simplified the material on information in games (Chapter 8). In particular, they provided an expanded description and more examples of preliminary exchanges in order to clarify the relationship between the alignment of interests and the possibility of reliable communication. In addition, we analyzed examples of signaling and screening at the beginning rather than at the end of the chapter, as was the case in the third edition, to convince students of the importance of this topic and to set the stage for the drier theory presented in the following sections.
Games in some of the applications of game theory discussed in later chapters were simple enough to be analyzed without an extensive game tree or payoff table. But this weakened the connection between the previous chapters, which outlined the methodological principles of game theory, and examples of the practical application of these principles. Therefore, we now show more inference tools in the context of their practical use.
We have expanded and improved the set of exercises. They, as in the third edition, are divided into two groups in each chapter - with and without solutions - and in most cases are presented in parallel: for each exercise with a solution there is a corresponding exercise without a solution, but with minor changes, which allows students to practice further. Readers can access solutions to the first group of exercises at books.wwnorton.com/studyspace/disciplines/economics.aspx?DiscId=6. Solutions to the second group of exercises will be provided to teachers using this textbook in their work: they will need to contact the publisher to gain access to the site for teachers. In each group of exercises with and without solutions, there are two types of exercises. Some provide the opportunity to repeat and practice the methods being studied, in others we guide the student step by step through the process of creating a model for analyzing a particular issue or problem from the perspective of game theory (in our opinion, these exercises have the greatest educational value). Such experience, gained through the analysis of exercises with solutions and reinforced through corresponding exercises without solutions, contributes to the development of strategic thinking skills in students.
B O Most of the other chapters have also been updated, improved, systematized and simplified. The most significant changes have been made to the chapters on topics such as the prisoners' dilemma (Chapter 10), collective action (Chapter 11), evolutionary games (Chapter 12), and voting (Chapter 15). We have excluded the last chapter of the third edition (Markets and Competition) because evidence shows that almost no one used it. If necessary, teachers can find it in the third edition of the textbook.