Before revision:  In this work, we study , for the first time, infinite-duration all-pay bidding games , and show that they exhibit the elegant{\em   mathematical properties of their first-price counterparts .
After revision:  In this work, we study , for the first time, infinite-duration all-pay bidding games , and show that they exhibit the elegant{\em objectives: we portray a complete picture for games played on strongly-connected graphs. We study both pure (deterministic) and mixed (probabilistic) strategies and completely characterize the optimal sure and almost-sure (with probability 1) payoffs that the players can respectively guarantee. We show that mean-payoff games under all-pay bidding exhibit the intriguing mathematical properties of their first-price counterparts .