game theory chapter 7 8 9 and 10 part 3

DYNAMIC GAMES OF COMPLETE INFORMATION

Chapter 7 preliminaries

As we have seen in a number of examples, the normal-form representation seems like a general method of putting formal structure on strategic situations. It allowed us to analyze a variety of games and reach some conclusions about the outcomes of strategic interaction when players are rational, as well as when there is common knowledge of rationality. Setting a game within the limits of our “actions, outcomes, and preferences” language seems, therefore, to have been a useful exercise. The simplicity and generality of the normal form notwithstanding, one of its drawbacks is its inability to capture games that unfold over time. That is, there is a sense of how players’ strategy sets correspond to what they can do, and of how the combination of their actions affects each other’s payoffs, but there is no way to represent situations in which the order of moves might be important. As an example, consider the familiar Battle of the Sexes game:

but with a slight modification. Imagine that Alex finishes work at 3:00 p.m. while Chris finishes work at 5:00 p.m. This gives Alex ample time to get to either the football game or the opera and then to call Chris at 4:45 p.m. and announce “I am here.” Chris then has to make a choice of where to go. Where should Chris go? If the choice is to the venue where Alex is waiting then Chris will get some payoff. (It would be 1 if Alex is at the opera and 2 if Alex is at the football game.) If Chris’s choice is to go to the other venue, then he will get 0. Hence a rational Chris should go to the same venue that Alex did. Anticipating this, a rational Alex ought to choose the opera, because then Alex gets 2 instead of 1 from football. As you can see, there is a fundamental difference between this example and the simultaneous-move Battle of the Sexes game. Here Chris makes a move after learning what Alex did, hence the difference is in what Chris knows when Chris makes a move. In other words, in the simultaneous-move Battle of the Sexes game neither player knows what the other is choosing, so each player conjectures a belief and plays a best response to this belief. Here, in contrast, when it’s his turn to make a move, Chris knows what Alex has done, and as a result the notion of conjecturing beliefs is moot.

Furthermore Alex knows,by common knowledge of rationality, that Chris will choose to follow Alex because it is Chris’s best response to do so. As a result Alex can get what Alex wants. This is a simple yet convincing example of the commonly used phrase “first-mover advantage.” By moving first, Alex gets to set the evening’s venue. Before we draw far-reaching conclusions about the generality of the first-mover advantage, consider the Matching Pennies game. If I were to play with you, no matter what my position would be (player 1 or 2), I would be happy to give you the opportunity to show your coin first. I am sure that you can see why! This chapter lays out a framework that allows us to represent formally such sequential strategic situations and apply strategic reasoning to these new representations. We then go a step further and introduce a solution concept that captures the important idea of sequential rationality, which is the general version of the dynamic reasoning offered in the Battle of the Sexes example. Earlier movers will take into account the rationality of players who move later in the game. This idea should resonate with the backward induction procedure introduced in Section 2.4.1.

7.4 Summary . In addition to the set of players, their possible actions, and their payoffs from outcomes, the extensive-form representation captures the order in which they play as well as what each player knows when it is his turn to move. Game trees provide a useful tool to describe extensive-form games. . If a player cannot distinguish between two or more nodes in a game tree then they belong to the same information set. Care has to be taken to specify correctly each player’s information sets. . In games of perfect information, every player knows exactly what transpired before each of his turns to move, so each information set is a singleton. If this is not the case then the players are playing a game of imperfect information. . A pure strategy defines a player’s deterministic plan of action at each of his information sets. A mixed strategy is a probability distribution over pure strategies, while a behavioral strategy is a plan of probability distributions over actions in every information set. . Every extensive-form game has a unique normal-form representation, but the reverse is not necessarily true.

chapter 8 Credibility and Sequential Rationality

There is therefore something unappealing about the logic of Nash equilibrium in extensive-form games. The concept asks only for players to act rationally on the equilibrium path given their beliefs about what will transpire both on and off the equilibrium path. Nevertheless it puts no restrictions on the beliefs of players off the equilibrium path, nor on how they should consider such beliefs. In fact the normal-form representation of a sequential game is not able to address such a requirement, which adds constraints on what we would tolerate as “rational behavior.” We would, however, expect rational players to play optimally in response to their beliefs whenever they are called to move. In the simple coordination game in Figure 8.1 this would require that player 2 commit to play p when it is his turn to move. Applying this reasoning to the sequential-move Battle of the Sexes game in Figure 7.11 suggests that of the three Nash equilibria of the game, two are unappealing. The equilibria (O, oo) and (F, ff ) have player 2 committing to a strategy that, despite being a best response to player 1’s strategy, would not have been optimal were player 1 to deviate from his strategy and cause the game to move off the equilibrium path. In what follows we will introduce a natural requirement that will result in more refined predictions for dynamic games. These will indeed rule out equilibria such as (E, e) in the coordination game and (O, oo) and (F, ff ) in the sequential-move Battle of the Sexes game.

8.4 Summary . Extensive-form games will often have some Nash equilibria that are not sequentially rational, yet we expect rational players to choose sequentially rational strategies. . In games of perfect information backward induction will result in sequentially rational Nash equilibria. If there are no two payoffs at terminal nodes that are the same then there will be a unique sequentially rational Nash equilibrium. . Subgame-perfect equilibrium is the more general construct of backward induction for games of imperfect information. . At least one of the Nash equilibria in a game will be a subgame-perfect equilibrium.

Chapter 9 Multistage Games

The normal-form game was a model of players choosing their actions simultaneously, without observing the moves of their opponents. The extensive form enriches the set of situations we can model to include those in which players learn something about the choices of other players who moved earlier, in which case a player can condition his behavior on the moves of other players in the game. The extensive-form games analyzed so far were characterized by having payoffs delayed until the game reached a terminal node that was associated with the game’s end. In reality dynamic play over time may be more complex, and it may not be correctly modeled by one “grand” game that unfolds over time with payoffs distributed at the end of the game. For instance, consider two firms that compete in one market, with the flow of profits that results from their behavior; after some time they choose to compete in another market, with a new flow of payoffs. Similarly a group of elected lawmakers may serve on a committee, on which their behavior and votes will affect the likelihood of their being reelected. Those who are reelected may sit on another committee together, with different choices and different consequences. The broad message of these examples is that players can play one game that is followed by another, or maybe even several other games and receive some payoffs after each one of the games in this sequence is played. In such cases two natural questions emerge. First, if the players are rational and forward looking, should they not view this sequence of games as one grand game? Second, if they do view these as one grand game, should we expect that their actions in the later stages will depend on the outcomes of earlier stages? In particular, will the players be destined to play a sequence of action profiles that are Nash equilibria in each stage-game, or will they be able to use future games to support behavior in the earlier stages that is not consistent with Nash equilibrium in those early stages? This chapter answers these questions. Before doing so, Sections 9.1–9.3 formally show how such multistage games can be modeled as extensive-form games so that we can use the tools developed in the previous chapter. It will be possible to answer the two questions just posed. First, the players should indeed anticipate future games and use them to create a richer environment for themselves. Second, they will benefit from using future play to create incentives that constrain their behavior in earlier stages. If players have the opportunity to condition future behavior on past outcomes, they may be able to create credible, self-enforcing incentives to follow behavior in earlier-stage games that would not be possible if there were no continuation games following. This 175 176 . Chapter 9 Multistage Games is the core insight of multistage games: if players can condition future behavior on past outcomes then this may lead to a richer set of self-enforcing outcomes. 9.1 Preliminaries We define a multistage game as a finite sequence of normal-form stage-games, in which each stage-game is an independent, well-defined game of complete but imperfect information (a simultaneous-move game). These stage-games are played sequentially by the same players, and the total payoffs from the sequence of games are evaluated using the sequence of outcomes in the games that are played. We adopt the convention that each game is played in a distinct period,so that game 1 is played in period t = 1, game 2 in period t = 2, and so on, up until period t = T, which will be the last stage in the game.1 We will also assume that, after each stage is completed, all the players observe the outcome of that stage, and that this information structure is common knowledge.2 In each stage-game, players have a set of actions from which they can choose, and the profiles of actions lead to payoffs for that specific game, which is then followed by another game, and another, until the sequence of games is over. A sequence of normal-form games of complete information with the same players is not too hard to conceive. Consider again the examples used earlier in this chapter, such as firms in common markets or lawmakers interacting on a sequence of committees.

9.6 Summary . Multistage games are defined by a series of normal-form stage-games that are played in sequence, in which players obtain payoffs after every stage-game and future payoffs are discounted. . Any sequence of stage-game Nash equilibrium play can be supported as a subgame-perfect equilibrium in the multistage game, regardless of the discount factor. . Players can use credible threats and promises in later stages to provide longterm incentives for short-term actions that may not be self-enforcing in the earlier-period stage-games. . Because future payoffs are discounted, the effectiveness of long-term incentives will depend on how patient the players are. . The set of outcomes that can be supported by a subgame-perfect equilibrium will often depend on the discount factor.

Chapter 10 10 Repeated Games

The previous chapter on multistage games demonstrated two important lessons. First, when players play a sequence of games over time, it will be to their benefit to use conditional strategies in later stage-games to support desirable behavior in early stage-games. More importantly the behavior that can be supported need not be a static best response in the early stage-game because the conditional strategies in later stage-games can act as a powerful incentive scheme to help players resist the short-run temptation of deviating from the proposed path of play. Second, and equally important, the future that the players face must be important enough to support these dynamic incentives as self-enforcing. Using so-called reward-and-punishment strategies to sustain static non-best-response behavior is possible only if the players do not discount the future too heavily. A special case of multistage games has received considerable attention over the years: the case of repeated games. A repeated game is simply a multistage game in which the same stage-game is being played at every stage. These games have been studied for two primary reasons. The first is that repeated games seem to capture many realistic settings. These include firms competing in the same market over long periods of time, politicians engaging in pork-barrel negotiations in session after session of a legislature, and workers on a team production line who perform some joint task day after day. The second reason is that repeating the same game over time results in a very convenient mathematical structure that makes the analysis somewhat simple and elegant. As such, social scientists were able to refine this analytical tool and provide a wide range of models that can be applied to understand different social phenomena. This chapter provides a glimpse into the analysis of repeated games and shows some of the extreme limits to what we can do with reward-and-punishment strategies.

10.7 Summary . If a stage-game that has a unique Nash equilibrium is repeated for finitely many periods then it will have a unique subgame-perfect equilibrium regardless of the discount factor’s value. . If a stage-game that has a unique Nash equilibrium is repeated infinitely often with a discount factor 0 <δ< 1, it will be possible to support behavior in each period that is not a Nash equilibrium of the one-shot stage-game. The “carrot” and “stick” incentives are created by bootstrapping the repetition of the stage-game’s unique equilibrium, which becomes a more potent threat as the discount factor approaches 1. . The folk theorem teaches us that as the discount factor approaches 1, the set of average payoffs that can be supported by a subgame-perfect equilibrium of the infinitely repeated game grows to a point that almost anything can happen. . Repeated games are useful frameworks to understand how people cooperate over time, how firms manage to collude in markets, and how reputations for good behavior are sustained over time even when short-run temptations are present.

chapter 11 Strategic Bargaining (this chapter is optional, bargaining is not part of the syllabus)

Of the many possible applications of strategic interaction among a small number of players, one ubiquitous situation surely comes to mind: bargaining. Examples are plentiful: a firm and a union bargaining over wages and benefits; a local municipality bargaining with a private provider over the terms of service; the head of a political party bargaining with other party members over campaign issues; and even the mundane bargaining that occurs between a buyer and a merchant at a street bazaar. In all of these examples, deals are struck by parties bargaining over terms, payments, and other aspects that all boil down to one basic fact: each party wants to get the best deal it can. How can we model and analyze bargaining through the lens of game theory? As demonstrated by the examples just mentioned, the issue is often about a surplus that has to be split among the parties, and it involves the parties making proposals, responding to them, and trying to settle on an agreement. Following some early work by Stahl (1972, 1977), Rubinstein (1982) offered a particular stylized model of ˚ bargaining that has become the standard in most applications. The simple framework considers two players that need to split a “pie” (representing the surplus from an agreement or the gains from trade). The pie is assumed to have a total value that is normalized to equal 1, and then the parties bargain on how to split the pie. In order to analyze this process as a strategic game, the bargaining process itself needs to be structured to include actions, outcomes that result from actions, and preferences over the outcomes, as with any other game we can imagine. Stahl suggested a ˚ prespecified procedure in which the game starts with player 1 making an offer to split the pie and player 2 either accepting or rejecting the offer. If the offer is accepted, the split proposed by player 1 is implemented, and the game ends. If player 2 rejects player 1’s offer, the players then switch roles: player 2 makes the offer and player 1 responds by accepting or rejecting the offer. The game can either continue this way until some exogenous deadline arrives or simply go on forever. Yet, as the saying goes, “time is money,” and delay ought to impose some loss on the players. There are two ways to impose this loss. First, there may be a fixed cost of progressing from one round to another, so that a constant piece of the pie is removed every time a rejection occurs. Alternatively we can impose discounting in the same way we did for repeated games: after every period of rejection, the total size of the pie is only a fraction of what it was before. This second approach is the one we will be following here.

11.5 Summary . Bargaining situations are common across many social settings. Stylized models of bargaining games can help shed light on the outcomes that are more likely to occur as a consequence of the bargaining protocol. . A key determinant of how players will share the surplus from an agreement is the discount factor. The higher the discount factor the more patient the players, and the surplus will be shared more equally among them. The bargaining protocol and agreement rules will have a significant impact on how the surplus is shared. In finite games with prespecified alternating moves, the last player to move has a last-mover strategic advantage, while the first player has an advantage owing to discounting. . When the game is not finite or when the role of proposer is determined randomly with equal probability, there is no last-mover advantage and only the first-mover advantage remains.


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