Lobbying is modeled as a first price auction, in which each lobbyist makes a "take it or leave it offer," consisting of a policy and a transfer. In contrast to a conventional auction, the politician (the seller) has preferences over policy, and the winning bid may impose externalities across the lobbyists (bidders). Pure strategy Nash equilibria exist and are completely characterized. In equilibrium, each lobbyists offers the "pairwise optimal" policy that maximizes the sum of utility for the lobbyist and politician. With two lobbyists, the winner is the constrained efficient lobbyist, i.e., the total utility of all players at her pairwise optimum exceeds the total utility at the pairwise optimum of the loser. With multiple lobbyists, there is an equilibrium in which a lobbyist wins if and only if the lobbyist is maximal with respect to an acyclic competitiveness relation. In the spatial model with quadratic utility, this implies that there is always an equilibrium in which the lobbyist furthest from the politician wins, and there is never an equilibrium in which the closest lobbyist wins. With two lobbyists, mixing does not meaningfully change the outcome of the game, but when there are multiple lobbyists, mixing allows the politician to do better: given any pure strategy equilibrium, there is a mixed strategy equilibrium that increases the politician's payoff.
John Duggan is a professor of political science and economics at the University of Rochester, and he is a research associate of the W. Allen Wallis Institute of Political Economy. He was director of the W. Allen Wallis Institute of Political Economy for the period 2002-2012, and he was co-managing editor for the journal Social Choice and Welfare for the period 2008-2015. He received his PhD in social science from the California Institute of Technology in 1995. His specializations are game theory, political economy, and social choice theory. His current work is on equilibrium existence in non-cooperative games, dynamic models of bargaining and elections, multi-dimensional spatial models of political competition, and informational aspects of voting and elections.