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Trees and decisions

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  • Carlos Alós-Ferrer

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  • Klaus Ritzberger

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Abstract

The traditional model of sequential decision making, for instance, in extensive form games, is a tree. Most texts define a tree as a connected directed graph without loops and a distinguished node, called the root. But an abstract graph is not a domain for decision theory. Decision theory perceives of acts as functions from states to consequences. Sequential decisions, accordingly, get conceptualized by mappings from sets of states to sets of consequences. Thus, the question arises whether a natural definition of a tree can be given, where nodes are sets of states. We show that, indeed, trees can be defined as specific collections of sets. Without loss of generality the elements of these sets can be interpreted as representing plays. Therefore, the elements can serve as states and consequences at the same time. Copyright Springer-Verlag Berlin/Heidelberg 2005

Suggested Citation

  • Carlos Alós-Ferrer & Klaus Ritzberger, 2005. "Trees and decisions," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 25(4), pages 763-798, June.
  • Handle: RePEc:spr:joecth:v:25:y:2005:i:4:p:763-798 DOI: 10.1007/s00199-004-0487-3
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    Cited by:

    1. Carlos Alós-Ferrer & Klaus Ritzberger, 2017. "Characterizing existence of equilibrium for large extensive form games: a necessity result," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), pages 407-430.
    2. repec:eee:gamebe:v:103:y:2017:i:c:p:19-29 is not listed on IDEAS
    3. J. Jude Kline & Shravan Luckraz, 2016. "Equivalence between graph-based and sequence-based extensive form games," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(1), pages 85-94, April.
    4. Joseph Greenberg & Sudheer Gupta & Xiao Luo, 2009. "Mutually acceptable courses of action," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 40(1), pages 91-112, July.
    5. repec:spr:jogath:v:46:y:2017:i:2:d:10.1007_s00182-016-0534-x is not listed on IDEAS
    6. Alós-Ferrer, Carlos & Kern, Johannes, 2015. "Repeated games in continuous time as extensive form games," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 34-57.
    7. Alós-Ferrer, Carlos & Ritzberger, Klaus, 2008. "Trees and extensive forms," Journal of Economic Theory, Elsevier, vol. 143(1), pages 216-250, November.
    8. Peter A. Streufert, 2015. "Concisely Specifying Choices in an Outcome-Set Form," UWO Department of Economics Working Papers 20152, University of Western Ontario, Department of Economics.
    9. Alós-Ferrer, Carlos & Ritzberger, Klaus, 2016. "Equilibrium existence for large perfect information games," Journal of Mathematical Economics, Elsevier, vol. 62(C), pages 5-18.

    More about this item

    Keywords

    Extensive form games; Sequential decisions; Trees.;

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D70 - Microeconomics - - Analysis of Collective Decision-Making - - - General

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