Choosing from an incomplete tournament Irem Bozbay Istanbul Bilgi University
Ipek Ozkal-Sanver Bilgi University
Remzi Sanver Istanbul Bilgi University
Abstract By an "incomplete tournament" we mean an asymmetric binary relation. We introduce and axiomatize solution concepts for incomplete tournaments and compare our results with those established for tournaments (complete and asymmetric binary relations) and weak tournaments (complete binary relations).
Submitted: March 25, 2008.
Choosing from an incomplete tournament I·rem Bozbay, I·pek Özkal-Sanver and M. Remzi Sanver Istanbul Bilgi University February, 2008
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Extended Abstract
Tournaments are complete and asymmetric binary relations over …nite sets. Tournaments and tournament solutions are exhaustively investigated in the literature. Complete and asymmetric binary relations are encountered oftenly in sports, collective decision making procedures, etc. Peris and Subiza (1999) introduce “weak tournaments”in which ties are allowed. In our work, we focus on the how to choose from asymmetric binary relations, as we call “incomplete tournaments”. Incomplete tournaments have broad real life implications. In many sports tournaments like tennis, players do not play with every other player, so, the resulting relation is not complete. The overall ranking of the players depends on the results of their matches, and also the strength of the players they play with. We adapt some solution concepts -top cycle of Schwartz (1972), Miller (1977); uncovered set of Miller (1980); minimal covering set of Dutta (1988)- established for tournaments to incomplete tournaments and axiomatize them. Let A be a …nite set of alternatives. We write for the set of asymmetric binary relations over A. Any T 2 is called an incomplete tournament. We say that x; y 2 A are comparable if xT y or yT x holds. An alternative x 2 A is said to be ”comparable in A with respect to T " if and only if there exists at least one y 2 A such that x:and y are comparable. For any x; y 2 A with neither xT y nor yT x; we write x y: Corresponding author:
[email protected]
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A trivial way of completing a binary relation is assuming indi¤erence betweeen the alternatives which are not comparable. The binary relation R with strict counterpart P and indi¤erence counterpart I is de…ned accordingly: 8x; y 2 A; xRy () not yT x 8x; y 2 A; xP y () xT y; 8x; y 2 A; xIy () x y Through this R; we obtain a weak tournament which was de…ned by Peris and Subiza (1999). We write for the set of weak tournaments. More formally, a weak tournament solution is a mapping F : 2A ! 2A such that F (T; X) X 8(T; X) 2 2A : What are the possible problems that may arise from assuming indi¤erence between not comparable alternatives? One of them is illustrated in the folowing example: Example 1.1 Let X = fx1 ; x2 ; ::::; x100 g: x1 T x 8x 2 X n fx100 g, xi T xj for 2 i < j; and x1 x100 : In this incomplete tournament, x1 and x100 are not comparable. Assuming that they are indi¤erent, we obtain a weak tournament R de…ned according to the de…nition above in which x1 P x; 8x 2 X n fx100 g, xi P xj for 2 i < j; and x1 Ix100 : The uncovered set of this weak tournament according to the covering relation by Peris and Subiza (1999)1 is U C(R; A) := fx1 ; x2 ; x100 g: If we imagine that this is a sports tournament, one of the winners is a player that is beaten by all of her opponents and has one game missing. Next, we propose another way of observing a complete tournament from any T 2 : For any couple (x; y) 2 A A; we say that a path P (x; y) from x to y is a sequence xh h=1;:::;H in X such that x1 = x; xh = y and xh T xh+1 for all h = 1; :::; H 1: The length of the shortest path from x to y is called P min (x; y): If there is no such sequence from x to y; then P min (x; y) = 1. An incomplete tournament solution is a mapping S : that S(T; X) X; 8(T; X) 2 2A : 1
yP w ) xP w : yIw ) xRw A ! 2 is the set of uncovered elements corresponding
x covers y in A i¤ xP y and 8w 2 A;
The uncovered set U C : to the de…nition.
2A
2A ! 2A such
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For any T 2
, any X 2 2A we will de…ne the following axioms:
An incomplete tournament solution S :
2A ! 2A satis…es
A1 if for any y 2 S(T; X); i¤ y 2 S(T; X) and x 2 = S(T; X) imply P min (y; x) < P min (x; y)) A2 if there exists no nonempty Y S(T; X) such that P min (x; y) < P min (y; x) 8x 2 Y , 8y 2 S(T; X) Y: A3 if given Y X with P min (y; x) < P min (x; y) for 8y 2 Y and 8x 2 X Y; then Y S(T; X):) Let X 2 2A . The top cycle choice correspondence T C : 2A ! 2A assigns the set T C(T; X) = fx 2 X : 8y 2 X there is an integer k and a sequence \x = z1 ; :::; zk = y" such that P min (zi ; zi+1 ) P min (zi+1 ; zi ) for all i 2 f1; :::; k 1gg to each T 2 : We have the following characterization result: An incomplete tournament solution S : and A3 if and only if S(T; X) = T C(T; X):
2A
! 2A satis…es A1; A2
We also present an axiomatization of solution concepts which adapt the uncovered set, Copeland rule (1951) and minimal covering set to the incomplete tournaments framework.
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References
Copeland A. (1951) A reasonable social welfare function. Mimeo, University of Michigan Seminar on the Applications of Mathematics to Social Sciences. Dutta B (1988) Covering sets and a new Condorcet choice correspondence. Journal of Economic Theory 44: 63-80. Miller, N.R. (1977) Graph theoretical approaches to the theory of voting Am. J. Polit. Sci. 21: 769-803. Miller, N.R. (1980) A new solution set for tournament and majority voting: further graph theoretical approaches to the theory of voting. Am. J. Polit. Sci. 24: 68-69. Peris and Subiza (1999) Condorcet choice correspondences for weak tournaments. Social Choice and Welfare 16: 217-231. Schwartz T (1972) Rationality and the myth of maximum. Nous 6:97-117. 3
