The Paradox of Vote Trading: Effects of Decision Rules and Voting Strategies on Externalities

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Abstract
In an article, “The Paradox of Vote Trading,” (APSR 67 [December, 1973]) William H. Riker and Steven J. Brams have argued that systematic logrolling among all members of a legislature produces a paradox: While each trade is individually rational, the effects of externalities offset the potential gains from exchanging votes and each voter finds himself worse off than he would have been by voting sincerely. We extend the results of Riker and Brams to a unanimity decision rule and find that a paradox of vote trading holds for that decision rule as well as for simple majority rule. Under a unanimity rule, however, trades which would be collectively rational (i.e., which would produce a Pareto optimal result) are not individually rational; the non-trader is the beneficiary under such a decision rule. Finally, we pose the question Riker and Brams suggested: Is the paradox of vote trading inescapable? Except under very restrictive conditions, we find that it is. However, given certain assumptions about the distributions of individual utilities, we present proofs of the necessary and sufficient conditions for the Pareto optimality of vote trading and argue that in actual legislative situations, when vote trading is Pareto optimal, learning behavior should serve to extricate the members from the paradox of vote trading.