Abstract
We analyze the Condorcet paradox within a strategic bargaining model with majority voting, exogenous recognition probabilities, and no discounting for the case with three players and three alternatives. Stationary subgame perfect equilibria (SSPE) exist whenever the geometric mean of the players’ risk coefficients, ratios of utility differences between alternatives, is at most one. SSPEs ensure agreement within finite expected time. For generic parameter values, SSPEs are unique and exclude Condorcet cycles. In an SSPE, at least two players propose their best alternative and at most one player proposes his middle alternative with positive probability. Players never reject best alternatives, may reject middle alternatives with positive probability, and reject worst alternatives. Recognition probabilities represent bargaining power and drive expected delay. Irrespective of utilities, no delay occurs for suitable distributions of bargaining power, whereas expected delay goes to infinity in the limit where one player holds all bargaining power. An increase in the recognition probability of a player may weaken his bargaining position. A player weakly improves his bargaining position when his risk coefficient decreases.
Highlights
Decisions on collective choice problems are often taken by means of majority voting, and the analysis of majority voting is an important topic in political economy
Condorcet winners may not exist and this gives rise to the Condorcet paradox in which any alternative can be reached from any other by a sequence of alternatives, where each alternative in the sequence beats the previous one by a pairwise majority vote as has been demonstrated in McKelvey (1976, 1979)
We show that across all parameter values seven equilibrium types are possible, three of which occur for a degenerate set of parameter values only, leaving four generic equilibrium types
Summary
Decisions on collective choice problems are often taken by means of majority voting, and the analysis of majority voting is an important topic in political economy. We study SSPE cycles in the sense of whether there is a positive probability that an equilibrium path can result in which all three alternatives have been proposed and rejected before some alternative is accepted Such SSPE cycles do not occur, though SSPE cycles are possible in degenerate cases. Restricting attention to the generic parameter values for which there is a unique SSPE, we find that decreasing a player’s risk coefficient either has no effect at all or leads to a higher probability of attaining his best alternative and a lower probability of attaining his worst alternative, so improves the player’s bargaining position. Banks and Duggan (2000) generalize the set-up of Baron and Ferejohn (1989) in several directions, and study collective decision making on a non-empty, compact and convex set of alternatives They include an analysis of the case where all discount factors are equal to one.
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