## Using Theorema in the Formalization of Theoretical Economics

Venue: | Proceedings CICM 2011, pages 58–73, 2011. + 11] [KW11] [Lan11] [Lib10 |

Citations: | 2 - 1 self |

### BibTeX

@INPROCEEDINGS{Kerber_usingtheorema,

author = {Manfred Kerber and Colin Rowat and Wolfgang Windsteiger},

title = {Using Theorema in the Formalization of Theoretical Economics},

booktitle = {Proceedings CICM 2011, pages 58–73, 2011. + 11] [KW11] [Lan11] [Lib10},

year = {}

}

### OpenURL

### Abstract

Abstract. Theoretical economics makes use of strict mathematical methods. For instance, games as introduced by von Neumann and Morgenstern allow for formal mathematical proofs for certain axiomatized economical situations. Such proofs can—at least in principle—also be carried through in formal systems such as Theorema. In this paper we describe experiments carried through using the Theorema system to prove theorems about a particular form of games called pillage games. Each pillage game formalizes a particular understanding of power. Analysis then attempts to derive the properties of solution sets (in particular, the core and stable set), asking about existence, uniqueness and characterization. Concretely we use Theorema to show properties previously proved on paper by two of the co-authors for pillage games with three agents. Of particular interest is some pseudo-code which summarizes the results previously shown. Since the computation involves infinite sets the pseudocode is in several ways non-computational. However, in the presence of appropriate lemmas, the pseudo-code has sufficient computational content that Theorema can compute stable sets (which are always finite). We have concretely demonstrated this for three different important power functions. 1

### Citations

1050 |
Theory of games and economic behavior
- Neumann, Morgenstem
- 1947
(Show Context)
Citation Context ...xplore and prove properties of stylized economic environments. Since the Second World War, one particularly important body of theory has been game theory, as introduced by von Neumann and Morgenstern =-=[17]-=- and successfully developed by John Nash. The game theory stemming from von Neumann and Morgenstern has become known as cooperative game theory; it allows abstraction from the details of how agents mi... |

129 | Proofs and Refutations - Lakatos - 1976 |

122 |
On the complexity of cooperative solution concepts
- Deng, Papadimitriou
- 1994
(Show Context)
Citation Context ...d partition function form) as boundary points. At the same time, even though each is defined over an uncountable domain, their structure has thus far been sufficient to avoid Deng and Papadimitriou’s =-=[5]-=- pessimistic conclusions that whether a solution even exists may be undecidable. Thus, pillage games provide a class of games for analysis that is simultaneously rich and tractable. To our knowledge, ... |

97 | Productive use of failure in inductive proof
- Ireland, Bundy
- 1996
(Show Context)
Citation Context ...uided. In most cases, formulating an appropriate lemma is then a straightforward exercise, so that even this can be automated. There is a literature on lemma speculation from failing proofs (see e.g. =-=[7]-=-); some mechanisms have been implemented in the Theorema system as well (see [1]). In this case study, these tools have not been employed. 3.3 Lemma 3: The Core The dominion D[Y, π, n] denotes the set... |

60 | Theorema: Towards Computer-Aided Mathematical Theory Exploration - Buchberger, Craciun, et al. |

57 |
2013) Gambit: Software Tools for Game Theory, Version 13.1.0, http://www.gambit-project.org
- McKelvey, McLennan, et al.
(Show Context)
Citation Context ...ples with n > 3. It is understandable that such problems occur since typically for any new axiom set humans have initially no or only limited intuition. This way it is easy to as4 For example, Gambit =-=[14]-=- solves a broad class of non-cooperative games. 2sume false theorems and to overlook cases in proofs. Proofs found in mathematics in general and theoretical economics in particular, can be viewed fro... |

53 |
The Theorema Project: A Progress Report
- Buchberger, Dupré, et al.
(Show Context)
Citation Context ...al definitions into one ‘Definition’-environment, e.g. one definition for the axioms (WC), (WR), and (SR), where one ‘bound’-statement would suffice for all three axioms. For more details we refer to =-=[1]-=-. 6 For full formalizations in Theorema together with proofs see also http://www.cs. bham.ac.uk/~mmk/economics/theorema. 5∀ (xi = yi) =⇒ (π[C, x] = π[C, y]) ] i∈C In order to prove this, we call the ... |

12 |
Pillage and property
- Jordan
- 2006
(Show Context)
Citation Context ...eory. 4 In the current work, we use Theorema [19] to formalize a particular cooperative game form, called pillage games, and to prove formally certain properties of them. Pillage games, introduced in =-=[8]-=-, form an uncountable set of cooperative games, taking the two best known classes of cooperative games (those in characteristic and partition function form) as boundary points. At the same time, even ... |

11 |
Theory Exploration Versus Theorem Proving
- Buchberger
(Show Context)
Citation Context ...e approach in this kind of theory exploration is to always prove all properties of interactions between available concepts before introducing a new concept. This approach is advocated for instance in =-=[3]-=-. In the concrete case, it would require proving many properties of permuted vectors and swaps before talking about anonymity. Theorema, however, also supports a much more goaloriented approach: in th... |

9 |
A game with no solution
- Lucas
(Show Context)
Citation Context ...e of the key concepts of the field, the so-called stable set (by them just called the “solution”), always existed in games in characteristic function form. This was subsequently demonstrated by Lucas =-=[12]-=- to be incorrect. 2. Nobel Prize winning economist and game theorist Maskin [13] claimed that certain properties of a game in partition function form extended from n = 3 to n > 3. However, de Clippel ... |

8 |
Social choice theory in HOL: Arrow and Gibbard-Satterthwaite
- Nipkow
(Show Context)
Citation Context ...k on other axiomatic proofs within economic theory which have been formalized. Arrow’s theorem in social choice has attracted the most attention, including studies by Wiedijk [18] using Mizar, Nipkow =-=[15]-=- using HOL, and Grandi and Endriss [6] using Prover9. Noncooperative game theory has also received attention, including by Vestergaard and co-authors [16] with Coq. The formalization of a particular g... |

5 |
The Inductive and Modal Proof Theory of Aumanns Theorem on Rationality
- Vestergaard, Lescanne, et al.
- 2006
(Show Context)
Citation Context ...g studies by Wiedijk [18] using Mizar, Nipkow [15] using HOL, and Grandi and Endriss [6] using Prover9. Noncooperative game theory has also received attention, including by Vestergaard and co-authors =-=[16]-=- with Coq. The formalization of a particular game form in economics can be interesting to computer science for at least two reasons, and to economics for at least one. For computer science, economics ... |

4 |
First-Order Logic Formalisation of Arrow’s Theorem
- Grandi, Endriss
- 2009
(Show Context)
Citation Context ...omic theory which have been formalized. Arrow’s theorem in social choice has attracted the most attention, including studies by Wiedijk [18] using Mizar, Nipkow [15] using HOL, and Grandi and Endriss =-=[6]-=- using Prover9. Noncooperative game theory has also received attention, including by Vestergaard and co-authors [16] with Coq. The formalization of a particular game form in economics can be interesti... |

3 | Formalizing Arrow’s theorem
- Wiedijk
- 2009
(Show Context)
Citation Context ...tive game. Related is work on other axiomatic proofs within economic theory which have been formalized. Arrow’s theorem in social choice has attracted the most attention, including studies by Wiedijk =-=[18]-=- using Mizar, Nipkow [15] using HOL, and Grandi and Endriss [6] using Prover9. Noncooperative game theory has also received attention, including by Vestergaard and co-authors [16] with Coq. The formal... |

3 | An automated prover for Zermelo-Fraenkel set theory in Theorema
- Windsteiger
(Show Context)
Citation Context ...ency of swapping, which is trivial but still automatically verified based on the definition of σ1,2. These proofs, and all that will follow, have been generated by the Theorema set theory prover, see =-=[20]-=-. Equipped with Lemma[“perm swap”] and Lemma[“ANswap”] in the knowledge base, a three page humanunderstandable proof of the two directions of Lemma 2 is obtained. Note, however that the introduction o... |

1 | coalitions and externalities: A comment on Maskin - Bargaining |

1 |
Stable Sets in Three Agent Pillage Games
- Kerber, Rowat
- 2009
(Show Context)
Citation Context ... general properties about pillage games (such as the finiteness of the stable set) and investigated the possibilities of three particular power functions for arbitrarily many agents. Kerber and Rowat =-=[10]-=- studied an infinite class of power functions for three agents. In particular they gave a complete characterization of the stable set for arbitrary power functions (with three agents) which satisfy th... |