Sequential Games

4 Key excerpts on "Sequential Games"

• 

  eBook - ePub

  Game Theory

  A Modeling Approach

  • Richard Alan Gillman, David Housman(Authors)
  • 2019(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 5

  Sequential Play

  Sequential Games model scenarios in which the players act sequentially, usually taking turns according to some rule. As with strategic games, each player has complete information about each other’s strategies and preferences. We introduce the basic solution concepts, techniques, and key theorems. Further, we build models in a variety of contexts demonstrating a range of options in the number of players and the turn-taking process.

  A play of a sequential game consists of a sequence of actions taken by the players. Thus, we can describe a general game in terms of these sequences. The game begins with an empty sequence, known as the empty history and ends with a sequence called a terminal history . Terminal histories determine the outcomes of the game, just like strategy profiles determine outcomes in strategic games. As we did in that chapter, we will often use the terms interchangeably. Subsequences of play that are obtained along the way are known as non-terminal histories . We use the phrasing “history” rather than sequence to place emphasis on the fact that when a player takes an action, the history of previous actions taken is known. Thus, formally, we have the following definition.

  Definition 5.0.1. A sequential game consists of the following:

  1.A set

  N = { 1 , 2 , … , n }

  of at least two players.

  2.A set O of terminal histories. We define a history to be an ordered list (possibly empty) of player actions; a terminal history is a history with no actions following it.

  3.A player function , P , which assigns a player to every non-terminal history.

  4.A rule stating that starting with the empty history, if a non-terminal history is reached, the player assigned to it selects an action to append to the current history.

  5.Utility functions

  u i

  : O → R

  that specify the preferences among terminal histories for each player

  i ∈ N

  .

  At every non-terminal history, the assigned player must choose an action. In order to describe a strategy for a player in a sequential game, then, we must identify an action for every possible non-terminal history to which the player was assigned. This differs from strategies in strategic games which simply identified the single action each player would take.

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  Economics, Game Theory and International Environmental Agreements

  The Ca' Foscari Lectures

  • Henry Tulkens(Author)
  • 2019(Publication Date)
  • WSPC

    (Publisher)

  The phenomena and the decision problems I am concerned with in these lectures having been expounded in the preceding lectures, I now enter into a toolbox that should allow one at later stages to come to grips seriously with these problems. The presentation is made at a certain level of abstraction, because the instrument is of that nature, given its intended generality. Therefore, no reference whatsoever to environmental facts, nor to economic concepts in this lecture. I strictly stick to the language of the theory.

  4.1Strategic games in general2

  In the sense used in these lectures, a game is a mathematical object. Among its many possible forms, the one being presented in this section, called the strategic form, is the most directly applicable to our project. Other forms, like the extensive form, or the coalitional function form, will be introduced or alluded to later when useful for the same purpose.

  Definition 4.1. A game in strategic form (or a strategic game) is a triplet (N, Σ, v) where:

  •N = {i : i = 1, 2, . . . , n} denotes a set of individuals (actors), called players,

  •Σ = Σ1 × × Σ

  n

  denotes a set joint actions, called strategies, where for each i = 1, 2, . . . , n, Σ

  i

  is the set of strategies σ

  i

  ∈ Σ

  i

  that player i has access to and can choose from,

  •v = (v1 , . . . , v

  n

  ) is a vector of payoffs of the players, where for each i, v

  i

  ∈ R is the payoff of player i.

  The words “player”, “strategy” and “payoff” in this definition deserve some explanatory comment. The first one is a remnant of the origin of the theory: in its full-fletched current developments the “players” under consideration are, as suggested earlier, individuals considered to be able to act freely on anything. Much more than just participants in a parlor game. “Strategies” are plain actions that the players have access to, with no necessary tactical or military connotation whatsoever: only the material feasibility of these actions for each individual player is at stake. The word “payoff”, finally, designates the way in which a player evaluates for himself what occurs in the interaction process under study, be it in monetary units or in more qualitative or abstract terms. Thus, the “mathematical object” designated as a “game” is a quite general way to describe who does what and why in well-defined interactive situations between a given set of individuals who act according to a well-defined objective.

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  Basic Mathematics for Economics, Business and Finance

  • EK Ummer(Author)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  strategic interdependence. Strategic interdependency is a situation where the actions of a player or a group of players affect, and are affected by, the actions of other players or other groups of players. We can now define game theory as a mathematical toolkit that helps make optimal decisions under conditions of strategic interdependence among the players who engage in a game.

  A game may involve two or more players. Players of a game may be individuals, firms, organizations, political parties, governments, etc. They make the strategic decisions in the context of games and, therefore, they are also called strategic decision-makers. A strategic decision or, in short, a strategy, constitutes a set of decisions, plans, or actions to be followed by each player during the course of the game. A strategy may be either a pure strategy (also called a deterministic strategy) or a mixed strategy (also called a randomized strategy). A pure strategy is a strategy in which a player makes a deterministic (or nonrandom or specific) action or decision. A mixed strategy is a strategy in which a player makes random choice (that is, choice based on probabilities) among possible actions or decisions (or pure strategies).

  In addition to players and strategic decisions, a game involves rules, outcomes, and payoffs. Rules of a game are predetermined and contain a set of information including who starts the game, what information the players possess, how the players start the game, etc. The result of the game represents the outcome of the game. The results may be expressed in cardinal units such as dollars or in ordinal units such as utility obtained. The payoff of a game is the net gain or value of the objective function associated with a possible outcome of a game. A player’s utility function or objective function is generally called that player’s payoff function. An optimal strategy in a game is the strategy that maximizes a player’s expected payoffs

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  A Guide to Modern Economics

  • Michael Bleaney, Prof David Greenaway(Authors)
  • 1996(Publication Date)
  • Routledge

    (Publisher)

  4 Non-zero-sum games may be co-operative or non-cooperative. They are co-operative when the agents can make binding agreements before acting, and are non-cooperative otherwise. For instance, two firms in duopoly may promise not to hurt each other, but there is no legal institution which could enforce this agreement; the game must be modelled as non-cooperative.

  The equilibrium concepts used in co-operative and non-cooperative games are very different. We shall concentrate only on the latter since they are appropriate for studying Schellingian ideas on strategic behaviour.

  Time

  If the agents meet only once and make decisions for a single period of time, the game is static in nature. In a broad sense a game is dynamic when time matters, either because a single-stage play is repeated over many periods (repeated game) or because the game unfolds over time in multi-stage plays. In solving multi-period complex games time can be treated as either a discrete or a continuous variable; the latter approach is technically more demanding. When the number of periods is limited, dynamic games are solved by backward induction, a technique developed in dynamic programming (Bellman 1957).

  For the most part, the situations dealt with in economics involve numerous meetings of the agents: firms operating in the same market know that they will have to meet again in many periods to come, and so do governments choosing their trade policies. Moreover, many economic variables, such as investment in capacity or advertising expenses, have clear effects on supply and demand conditions which will prevail in the future. All these features of economic life explain why dynamic games constitute the proper framework for studying strategic behaviour in economics.

  Information

  A crucial step in the description of a game is the specification of the structure of information for the different players. In most applications of game theory to economics it seems natural to assume that some information is private; for instance, each firm knows its own cost function but not the other players’ costs. These informational asymmetries give rise to very rich possibilities of strategic behaviour such as bluffing or reputation-building.

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