We consider two players, starting with m and n units, respectively. In each round, the winner is decided with probability proportional to each player's fortune, and the opponent loses one unit. We prove an explicit formula for the probability p(m, n) that the first player wins. When m ~ Nx0, n ~ Ny0, we prove the fluid limit as N → ∞. When x0 = y0, z → p(N, N + z√N) converges to the standard normal cumulative distribution function and the difference in fortunes scales diffusively. The exact limit of the time of ruin τN is established as (T - τN) ~ N-βW1/β, β = ¼, T = x0 + y0. Modulo a constant, W ~ χ21(z02 / T2).

Numerous archival documents show how the suspension of payments by Philip II, in September 1575, on the contracts with Genoese bankers (asientos) induced a freeze of the domestic credit market in Castile through the bankers’ intermediation for asientos and the credit interconnections. Commercial fairs stopped, banks failed and trade suffered while the king granted legal protection to the Genoese bankers. The evidence strikingly confirms that by his strategy, Philip II was able to remove the de facto ceiling on the domestic debt (juros) imposed by the fixed revenue commitment of the Castilian cities in the Cortes. The agreement with the bankers was signed in December 1577 immediately after the cities had agreed to the doubling of their commitment.

We consider stochastic replicator processes for games that are composed of finitely many trials. Several general results on the relation between Nash equilibria and the long-run behaviour of the stochastic processes are proved. In particular, a sufficient condition is given for almost sure convergence to a state where everyone plays in every trial a strict Nash equilibrium. The results are applied to multiple-trial conflicts based on wars of attrition and on sperm competition games with fair raffles, respectively.

We study an auction mechanism combining the war of attrition and the all-pay auction under the assumption that buyers’ signals are affiliated. We provide a sufficient condition which guarantees the existence of a symmetric, increasing equilibrium bidding strategy, but rules out full extraction of buyers’ surplus by the seller. We develop revenue comparison between the combined mechanism, the war of attrition and the all-pay auction.