Dice games and Arrows theorem

We observe that non-transitivity is not the most general way that Arrow’s impossibility theorem is re‡ected in dice games. Non-transitivity in dice games is a well-known phenomenon. It was described in several Scienti…c American columns in the 1970s, beginning with [3], and has since become a popular topic, discussed in articles like [4, 5] and textbooks on elementary mathematics and probability. In many discussions, non-transitivity of dice is mentioned in connection with voting paradoxes like Arrow’s impossibility theorem [1], which states that transitivity and several other seemingly natural conditions cannot all be satis…ed by any mechanism that might be used to determine the outcome of elections involving three or more candidates. Our purpose here is to observe that non-transitivity is not actually the most general way that dice games illustrate Arrow’s theorem: dice games connected with elections involving three or more candidates (or more generally, dice games connected with elections in which voters express three or more levels of preference) violate independence of irrelevant alternatives (IIA), but only dice games connected with elections involving four or more candidates can be non-transitive. If m is a positive integer then an m-sided generalized die with integer labels is simply a list X = (x1; :::; xm) of integers; for convenience’ sake we presume that x1 x2 ::: xm. For integers a b let D(a; b) denote the collection of all such dice with a x1 ::: xm b. If X = (x1; :::; xm); Y = (y1; ::; yn) 2 D(a; b) then we call jf(i; j)jxi > yjgj jf(i; j)jxi < yjgj the win-loss di¤erence of X against Y . X is stronger than Y if this di¤erence is positive, and Y is stronger if the di¤erence is negative; if the win-loss di¤erence is 0 thenX and Y are tied. This relation re‡ects the natural game in which a “roll”consists of picking one of the xi at random with probability 1=m, and picking one of the yj at random with probability 1=n; X wins that roll if xi > yj . Note that if b a = d c then D(a; b) is isomorphic toD(c; d) under the map (x1; :::; xm) 7! (x1+c a; :::; xm+c a). Arrow’s theorem [1] applies to elections in which each voter expresses a preference order of the candidates. A preference order is a re‡exive, transitive relation which is complete (no two candidates are incomparable); a voter may express identical levels of preference for some candidates. Observe that a preference order does not specify how strongly a voter prefers one candidate over another; see [1, 2] for discussions of the di¤erence between ordinal and cardinal measures of preference. The preference order of an individual voter is equivalently expressed by assigning each candidate the rating r = jfcandidates whom that voter does not strictly prefer to that candidategj. The ratings assigned to a given candidate by the various voters determine an element of D(1; c), where c is the number of candidates. If we allow voters to decide to abstain from assessing certain candidates, the dice may not all be the same size. Although the stronger relation on D(1; c) is a natural way to judge a dice game, it is rather unnatural as a vote-counting scheme: candidate X is stronger than candidate Y if, when one among the m voters who assessed X is chosen randomly with probability 1=m and one among the n voters who assessed Y is chosen randomly with probability 1=n, it is more likely than not that the …rst voter’s rating of candidate X is at least as large as the second voter’s rating of candidate Y . A crucial feature of the representation of elections with dice is commonly neglected. When we restrict our attention to an election involving a subset of the original set of candidates, the ballots pertinent to this “sub-election” are obtained by restricting the voters’preference orders to that subset; in general, restricting the preference orders will change the dice associated with the candidates. Example 1. Consider the Condorcet triple.