In many contests in economic and political life, both all-pay and winner-pay expenditures matter for winning. This paper studies such hybrid contests under symmetry and asymmetry. The symmetric model assumes very little structure but yields a simple closed-form solution. More contestants tend to lead to substitution toward winner-pay investments, and total expenditures are always lower than in the corresponding all-pay contest. With a biased decision process and two contestants, the favored contestant wins with a higher likelihood, chooses less winner-pay investments, and contributes more to total expenditures. An endogenous bias that maximizes total expenditures disfavors the high-valuation contestant but still makes her the more likely one to win.
Lagerlöf, Johan N M.
"Hybrid All-Pay and Winner-Pay Contests."
American Economic Journal: Microeconomics,
Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior
Asymmetric and Private Information; Mechanism Design