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Index
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Book:
- Game Theory
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- 31 May 2012, pp 441-448
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3 - Dominant strategies
- from Part II - Basic solution concepts for strategic form games
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Game Theory
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- 05 August 2012
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- 31 May 2012, pp 23-41
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Foreword
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- By Aviad Heifetz, The Open University of Israel
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Book:
- Game Theory
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- 31 May 2012, pp ix-xii
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Summary
FOREWORD
Game theory is concerned with strategic interaction among several decision-makers. In a strategic encounter of this kind, each player is aware of the fact that her actions affect the well-being of the other players, just as their actions affect hers. Game theory analyzes how these mutual influences channel the players’ decisions and lead to the ultimate outcome. Since the basic theoretical foundations of game theory were laid in the mid twentieth-century, numerous applications have been found for it in economics and management, as well as in political science, anthropology, sociology, biology and computer science.
It is important to realize that game-theoretic tools do not provide off-the-shelf solutions for predicting players’ behavior in a complex, real-life situation. In the social sciences, y compris in game theory, every model distils only a small number of critical aspects out of the vast plethora of dimensions characterizing a given situation. And it is exclusively in light of those aspects that it proceeds to analyze the situation, using highly stylized hypotheses regarding a whole host of other aspects. The purpose of the model is to provide a framework for systematic thinking about the complicated situation. Insights gained while analyzing the model may enable one to think more intelligently and profoundly about an actual, realistic situation. In this way, game theoretic models have shed light on the modus operandi of many economic and political mechanisms; and insights gained from such models have made a substantial contribution towards a more intelligent design of such mechanisms – incentives for workers in firms, financial markets and auctions for a large variety of assets, policies for diminishing air pollution, voting and election systems, and numerous other types of mechanisms, institutions and organizations.
22 - Moves of nature
- from Part VI - Dynamic games
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Book:
- Game Theory
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- 31 May 2012, pp 366-382
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Summary
So far, we have dealt with extensive form games, in which a given action profile by the players at a particular node always and unequivocally defines the next node to which this action profile leads. However, many strategic situations exist in which the development of the game does not depend solely on the actions of the players, and a certain randomness prevails over which the players have no control – either severally or jointly.
This randomness may be modeled with the aid of moves of nature at a chance node. This is a node on the game tree from which a number of branches divide; however, as distinct from nodes of the type we have dealt with hitherto, there are, at this node, no players who have to choose between the branches. Instead, there is a predefined probability at which each of the branches will be chosen. A node of this type may also be thought of as a node at which an imaginary player – “nature” – chooses how to act. Here, however, the probability of nature’s choice of each branch is given beforehand and is not a result of a conscious and mediated, intelligent choice. The “choice” of nature differs from the choices of the other players in that it is random and is not restricted to a definite choice of one of the branches.
When moves of nature are part of the game tree, the players’ strategies do not determine one unique path on the tree that leads to a particular leaf in a deterministic fashion. At every chance node, the game path splits into a number of possible continuations, in accordance with the probabilities dictated by the move of nature at that node; if the game has a number of chance nodes, the results of the lotteries at the various nodes are independent of one another. As a result, the players’ strategies determine a probability distribution over the leaves on the tree.
In the following sections we will analyze several extensive form games involving moves of nature.
17 - Global games
- from Part V - Advanced topics in strategic form games
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Game Theory
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- 31 May 2012, pp 284-300
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Summary
Equilibrium selection criteria
As we have seen, many games have more than one Nash equilibrium. In such games, additional criteria are needed in order to reach a more accurate prediction as to how the players will choose to play. Every such criterion is a selection criterion for choosing among the equilibria in the game.
In preceding chapters, we have already dealt with a number of such criteria. First, we saw examples of games in which, at some of the equilibria, some or all of the players choose weakly dominated strategies. In the Divvying up the Jackpot game, for example, we argued that it is natural to focus on equilibria at which the players do not choose such strategies. This is one possible criterion for excluding some of the equilibria in certain games.
We later discussed equilibria that are focal point equilibria – either because the payoff profile in such equilibria is more symmetric than in other equilibria, or because they are conspicuous by reason of external or historic causes that find expression not in the payoffs themselves but rather in the verbal description of the game.
In Chapter 9, we discussed two additional selection criteria.We defined when one equilibrium is more efficient than another. According to this criterion, if one of the equilibria in the game is more efficient than all the rest, the players will focus on that equilibrium. Moreover, if there is an equilibrium that is less efficient than another equilibrium, the players will not focus on the less efficient equilibrium.
The efficiency criterion does not always dovetail with another criterion we have examined – that of risk dominance. It is in games with multiple equilibria that the players are actually liable to be unsure as to how their rivals will play, and they may therefore choose strategies that appear to them as more secure on average. In a symmetric game of two players, they will hence play a risk-dominant equilibrium, which is not always the most efficient equilibrium.
In this chapter, we will present another approach for choosing among equilibria, one that is especially well suited to choosing between equilibria in coordination games.
6 - Nash equilibrium
- from Part II - Basic solution concepts for strategic form games
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Game Theory
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- 31 May 2012, pp 65-84
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Part VII - Repeated games
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Game Theory
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- 31 May 2012, pp 383-384
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Summary
INTRODUCTION
A repeated game is an extensive form game with an infinite horizon, in which the same normal form game is played again and again indefinitely. While the game repeats itself irrespective of the history of play, the players’ strategies may very well depend on the history of play. Along each infinite history of repetitions of the game, the payoff to each player is the discounted sum of her stage payoffs, with some discount factor between 0 and 1. The discounting is interpreted as either (1) the extent to which players prefer current payments over future ones, or (2) the probability that the game will not be halted but will actually continue into the future, or (3) as resulting from a monetary interest rate applying to saving of accumulated payments or to loans on account of future payments.
In Chapter 23 we study the infinitely repeated Prisoner’s Dilemma. Despite the fact that defection is a dominant strategy in the one-shot game, we show that the infinitely repeated Prisoner’s Dilemma has subgame perfect equilibrium strategies in which the players cooperate in all or at least some of the rounds. Such equilibrium strategies call the players to cooperate, and involve a threat of a “punishment phase” of defection for several rounds by the player who suffered from the initial unexpected defection if and when unexpected defection has indeed occurred. At equilibrium, the punishment phase is long enough so as to foreshadow the short-term gains that the defector could potentially achieve by surprising his opponent, and hence no unexpected defection actually occurs on the equilibrium path. The higher the discount rate, the shorter the punishment phase (from which the punisher might suffer as well) needed to induce the equilibrium cooperation.
Frontmatter
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Game Theory
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- 31 May 2012, pp i-vi
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24 - Games with an unbounded horizon
- from Part VII - Repeated games
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Book:
- Game Theory
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- 31 May 2012, pp 413-440
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Summary
In the preceding chapter, we discussed the repeated Prisoner’s Dilemma. In this game, the extent to which a player can “punish” or “reward” her rival in any round is fixed and predetermined. What happens in repeated games in which this extent is modifiable? How does this affect the set of equilibria in the game? We will examine this question in the following example.
Efficiency wages
According to the competitive economics model, in a perfect and frictionless market there should be no unemployment: if the supply of labor is greater than the demand for labor on the part of employers, workers will be prepared to work even at a lower wage – a wage at which employers will find it profitable to hire additional hands. The process of decrease in salaries will continue until the demand for employees equals the supply of labor.
In practice, however, even in competitive markets such as that of the United States, unemployment levels typically do not fall below 4–5 percent. One possible reason for such unemployment is the process of job search on the part of the unemployed, and the search for workers by potential employers. We discussed the modeling of job search and unemployment in Chapter 9.
We will now turn to examine another possible cause for the existence of a minimal level of unemployment, one that is related to the ongoing and repeated interaction between employers and employees.
Part IV - Uncertainty and mixed strategies
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Book:
- Game Theory
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- 31 May 2012, pp 157-160
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Summary
INTRODUCTION
In Chapter 10 we start addressing the issue of uncertainty. When a player is unsure what strategies his rivals will choose, we will assume that the player attaches a probability to each of the possible choice combinations. Each choice of a strategy of his own then defines the probability with which each profile of all the players’ choices would be realized. Overall, each of the player’s strategies defines a lottery over the strategy profiles in the game.
In order to decide which strategy to choose, the player then has to figure which of the lotteries induced by his strategies he prefers. We will assume that the player’s preference over these lotteries is expressed by the expected utility accrued by this lottery – the weighted average of his utilities from his choice and the others’ choices, weighted by the probabilities that he ascribes to the other players’ choices. This assumption means that the utility levels now have a cardinal (rather than just ordinal) interpretation. Preferences over lotteries which can be represented by an expected utility over outcomes are named after von Neumann and Morgenstern, who isolated four axioms on the preference relation which obtain if and only if an expected-utility representation of the preferences is feasible.
These axioms do not always obtain. We bring in the example of the Allais Paradox for preferences that seem “reasonable” but which nevertheless cannot be represented by an expected utility.
16 - Games and evolution
- from Part V - Advanced topics in strategic form games
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Book:
- Game Theory
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- 31 May 2012, pp 259-283
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Summary
The solution concepts we have dealt with so far were based on the assumption that the players taking part in the game are rational. Is game theory useful only under the assumption of rationality?
It may perhaps be surprising to discover that the answer to this question is negative. In this chapter, we will replace the assumption of rationality by which the players try to maximize their payoff with an assumption that is the very antithesis of rationality. We will assume that every player is devoid of the capacity to choose, being pre-programmed to play a unique particular strategy.
How can a theory of strategic choice be relevant for players who make no choice? Despite the apparent contradiction, we will see how the tools of game theory can serve for analyzing the evolution of characteristic properties or types of individuals in large populations. The fascinating connection between game theory and biology was explicitly proposed by Maynard Smith and Price (1973) and was later developed in the classic book of Maynard Smith (1982).
In population games, a player’s strategy is its type. In an animal population, the player is an individual in the population and its type is a genetic property that it bears – its genotype. This is a property that it inherits from its parents and bequeaths to its offspring. This idea can also be extended to a discussion of social and cultural norms in human societies: an individual’s type may be a social norm that the individual has assimilated and internalized in the course of the socialization process and which she will also bequeath to her children.
In the games we dealt with in the preceding chapters, the participants in the strategic encounter always constituted a constant (and usually small) set of players, and the encounter between them was a one-shot non-recurring interaction. In population games, by contrast, the encounters take place over and over again between pairs of individuals chosen in each encounter by random pairwise matching.
In any random encounter in the course of a population game, the individuals do not perform any process of choice; the payoff of each individual is determined by the profile of the types of individuals in the encounter. What, then, does this payoff express? If the individual makes no choice, what significance is there to the fact that the individual would have obtained a higher payoff had it been of a different type?
Game Theory
- Interactive Strategies in Economics and Management
- Aviad Heifetz
- Translated by Judith Yalon-Fortus
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- Published online:
- 05 August 2012
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- 31 May 2012
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Game theory is concerned with strategic interaction among several decision-makers. In such strategic encounters, all players are aware of the fact that their actions affect the other players. Game theory analyzes how these strategic, interactive considerations may affect the players' decisions and influence the final outcome. This textbook focuses on applications of complete-information games in economics and management, as well as in other fields such as political science, law and biology. It guides students through the fundamentals of game theory by letting examples lead the way to the concepts needed to solve them. It provides opportunities for self-study and self-testing through an extensive pedagogical apparatus of examples, questions and answers. The book also includes more advanced material suitable as a basis for seminar papers or elective topics, including rationalizability, stability of equilibria (with discrete-time dynamics), games and evolution, equilibrium selection and global games.
8 - Concentrated markets
- from Part III - Prominent classes of strategic form games
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Book:
- Game Theory
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- 31 May 2012, pp 118-136
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Summary
One of the most important applications of game theory in economics is that of analyzing the behavior of commercial firms. When production activity in a particular sector is carried out by a small number of firms, the situation is one of oligopoly or concentrated markets.
In some sectors, the prices set by the various firms for their products are either identical or very nearly so. Dairy manufacturers, for example, typically sell a quart of milk at the same or almost the same price. In the dairy market, therefore, firms compete over shelf space in the supermarkets and grocery stores, and, consequently, over the quantities sold to consumers. This is a typical example of quantity competition.
In other sectors, the quantity of units offered to customers is a constant, and firms compete primarily by means of the prices they charge consumers. For example, the flights operated by the various airlines on each route are scheduled well in advance, and the number of seats per airplane of a particular type is constant. Therefore, on short notice, airlines can neither increase nor decrease the number of seats offered to customers. They can, however, compete with each other by making changes in ticket prices and by launching various sales specials. This, therefore, is a typical example of price competition.
The various firms often differentiate products which are by and large similar to one another. For example, dairy manufacturers produce yoghurt treats which are similar but not identical. In this case we say that the firms’ products are differentiated.
If every firm has already reached agreement with the major supermarket chains as regards the shelf space that it is to be allocated for its yoghurt treats, then it remains only for each firm to quote a price for its product. In this case, we say that the firms are engaged in price competition with differentiated products.
But in certain periods, the main decision variable of yoghurt treat manufacturers will not necessarily be that of price. For example, once a new yoghurt treat has been launched, the firm may not wish to make frequent price changes so as not to confuse its customers. The firm may, however, reopen negotiations with the supermarket chains over the quantities of the product they are to purchase and display on their shelves. In this state of affairs, we say the firms engage in quantity competition with differentiated products.
13 - Mixed strategies in general games
- from Part IV - Uncertainty and mixed strategies
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Book:
- Game Theory
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- 31 May 2012, pp 211-234
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Summary
In Chapter 11, we defined mixed strategies in games in which every player has two pure strategies. In all the examples cited in Chapter 11, only two players took part in the game.
We will begin this chapter with an example of a game with many players in which there is a Nash equilibrium in mixed strategies. In the game in this example, every player has two strategies. In the second part of this chapter, we will extend the definition of mixed strategies to games in which every player has more than two pure strategies, and explore examples of such games.
The Volunteer’s Dilemma
Consider a group of honest people who witness the perpetration of a crime. Each of these people may prefer that one of the other people in the group come to the aid of the victim or call the police, because volunteering to do so is bothersome and might put the volunteer at risk. What are the Nash equilibria in this sort of social situation, and what are their properties? How do they depend on the number of witnesses?
This question was analyzed by Diekmann (1985). Assume that each of the witnesses, i = 1, . . ., n, has zero utility if nobody volunteers. If at least one of them volunteers to act on behalf of the victim, those who did not volunteer have a positive utility V, expressing their satisfaction with the assistance rendered to the victim. Each of the volunteers has a utility V – C. The term C expresses the effort or the risk experienced by the volunteer. We will assume that V – C > 0, i.e. that a unique witness to the crime would prefer to volunteer rather than to stand aside.
Part II - Basic solution concepts for strategic form games
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Book:
- Game Theory
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- 31 May 2012, pp 19-22
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Summary
INTRODUCTION
In Part II we will explore the basic solution concepts for strategic form games. The strongest solution concept of all, introduced in Chapter 3, is that of strongly dominant strategies. A strongly dominant strategy is one which is strictly preferable for the player irrespective of the choices of the other players.
We will see, though, that – somewhat surprisingly – even when all players have strongly dominant strategies, the outcome of the strategic interaction might be inefficient, and all players wish they could have coordinated to play a different strategy profile. The prime examples of this phenomenon are the two-player games of the Prisoner’s Dilemma type, called also “social dilemma” games.
A somewhat weaker solution concept, also discussed in Chapter 2, is that of weakly dominant strategies. A strategy is weakly dominant if, irrespective of the choices of the other players, the strategy is never inferior to any other strategy the player could have chosen, and for some strategy profile of the other players it is strictly better than all the alternative strategies available to the player. When each player has a weakly dominant strategy, the game is said to be solvable by weak domination.
A second price auction, which is akin (though not identical) to the one carried out in eBay and other online auctions, is an example of a game in which each bidder has a weakly dominant strategy – namely to bid the maximum amount she is willing to pay for the auctioned good. Upon winning, she will be paying only the second-highest bid, and hence any other bidding strategy is weakly dominated: bidding more than one’s maximum willingness to pay might lead to winning but paying more than this maximum, while bidding less than this maximum might lead to losing an opportunity to win the auction at a profitable price.
2 - Representing strategic interactions with games
- from Part I - Strategic interactions as games
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Book:
- Game Theory
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- 31 May 2012, pp 8-18
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Summary
In this chapter we will examine several examples that can be analyzed using game-theoretic tools. These examples will help to illustrate the considerations involved when social interactions or confrontations are represented by strategic form games. We will also discover what aspects cannot be represented by a strategic form game, and find out in what ways the game concept needs to be extended so as to realize more appropriate representations.
The background to the Six Day War
The Six Day War of June 5–10, 1967, between Israel and its neighboring Arab states Egypt, Jordan and Syria, was a key event in the evolution of the conflict in the Middle East. The strategic dilemmas faced by the belligerents constitute a prime example for game-theoretic analysis.
After Israel declared its independence on May 14, 1948, in accordance with the United Nations resolution from November 1947, its borders with Egypt, Jordan, Syria, and Lebanon were established via a war which lasted until March 1949. In 1956, in response to terrorist infiltrations from the Sinai Peninsula, Israel captured it from Egypt, but withdrew under international pressure and guarantees for shipping rights in the Red Sea, from the port of Eilat via the Straits of Tiran.
15 - Stability of equilibria
- from Part V - Advanced topics in strategic form games
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Book:
- Game Theory
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- 05 August 2012
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- 31 May 2012, pp 246-258
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Summary
At a Nash equilibrium a balance is struck between the players’ strategies: each player chooses a strategy that is optimal for her, given the strategies chosen by the other players. The equilibrium concept does not deal with the question of how such a balance is created, or what will happen if it is upset. In other words, the equilibrium is a static concept and does not address the question of what dynamics (if any) could possibly cause the players to choose, or to gradually approach, the equilibrium strategies.
A large number of dynamic processes are conceivable in which, over time, each player repeatedly updates her choice, and in so doing studies the moves of the other players and the payoffs she is getting. Of course, different processes will correspond to different assumptions concerning the players’ degree of sophistication, the information available to them, the memory resources and computational capability at their disposal, and so on. Accordingly, an important branch of modern game theory is called Learning in Games and this topic is one of the frontiers of game-theoretic research.
We will now proceed to describe two key types of updating processes.
Updating processes
Eductive processes
In eductive processes, the game is played one single time. Before it starts, each player mulls over various possible strategy profiles that both she and the other players may adopt, and in an iterative process progressively narrows down the possibilities that appear to her as reasonable.
In preceding chapters, we have already encountered two such types of eductive processes:
iterative elimination of strongly dominated strategies;
iterative elimination of weakly dominated strategies.
We have also seen that in certain games (such as Divvying up the Jackpot) the eductive process of iterative elimination of weakly dominated strategies can lead the players to exclude certain Nash equilibria.
14 - Rationalizable strategies
- from Part V - Advanced topics in strategic form games
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Book:
- Game Theory
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- 31 May 2012, pp 239-245
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20 - Commitment
- from Part VI - Dynamic games
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Game Theory
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- 31 May 2012, pp 333-352
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Summary
Commitment to action
In section 19.3.1, we described the Stackelberg model, in which one firm is the first to decide what quantity to produce, while the other firm decides what quantity it will produce only after having observed how much the first firm has produced. We saw that at the unique subgame perfect equilibrium of the game, the leading firm effectively chooses the combination of quantities that will maximize its profits from among all the combinations of quantities on its rival’s reaction curve. But in this combination of quantities, the quantity that the first firm produces does not constitute a best reply to the quantity that the second firm ultimately produces: if, after the second firm has chosen its strategy at a Stackelberg equilibrium, the first firm were to get a chance to change its mind and produce a smaller quantity than it had already produced, then it would certainly prefer to do so. Thus, the leading firm’s advantage is reflected in the fact that by its action, it can commit and bind itself to an action that at the end of the day is not necessarily optimal for it. The leading firm’s loss redounds to its advantage, since its commitment to an “over-aggressive” strategy leads its rival to behave “submissively.”
In the decision problem of a unique decision maker, the decision maker can derive no advantage from deciding to adopt an action which will prove, in retrospect, to have been suboptimal for her. One of the most important insights of game theory is that in strategic situations, a player’s ability to commit to her strategy – a strategy that is not necessarily optimal, given the reactions of her rivals – can secure her an important advantage.
7 - Cooperation and conflict, strategic complements and substitutes
- from Part III - Prominent classes of strategic form games
- Aviad Heifetz, Open University of Israel
- Translated by Judith Yalon-Fortus
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- Book:
- Game Theory
- Published online:
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- 31 May 2012, pp 89-117
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Summary
In the preceding chapters, we described a variety of games possessing various attributes, yet some of those games resembled one another in various ways. In Chapters 1 and 2 we described three different games of the “social dilemma” type, and we explained in what sense the three games are equivalent. More generally, finding common attributes for various games will enable us to catalog the games into categories, and to identify the forces and tensions that affect the equilibrium in a given category of games.
In this chapter, we take our first step in this cataloging process, by identifying two game attributes. The first attribute for differentiating between games is natural and simple. This attribute will answer the question: is the strategic interaction between the players one of cooperation or of conflict? We say that the strategic interaction is one of cooperation if every player always prefers that the other player or players intensify their actions. We will say that the strategic interaction is one of conflict if every player always prefers that the other player or players moderate their actions. The distinction between games of cooperation and games of conflict is accordingly relevant when the strategies of each player are completely ordered by their intensity or strength, from the most moderate to the most intensive strategy. An order relation of this kind is, in fact, definable in each of the examples we have encountered so far.
The second attribute we will present in this chapter, for the purpose of distinguishing between games, is more complex. It, too, is applicable only in games in which the strategies of each player can be ordered by their intensity. It will answer the question: will the player wish to intensify her actions when the other players intensify their actions? If every player will wish to intensify her action in response to the intensification of the activity of the other players or of at least some of them, we say that the game is one of strategic complements. If every player will wish to moderate her action in response to the intensification of the activity of the other players or of at least some of them, we say that the game is one of strategic substitutes.