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L\"ob's theorem and G\"odel's theorems make predictions about the behavior ofsystems capable of self-reference with unbounded computational resources withwhich to write and evaluate proofs. However, in the real world, systems capableof self-reference will have limited memory and processing speed, so in thispaper we introduce an effective version of L\"ob's theorem which is applicablegiven such bounded resources. These results have powerful implications for thegame theory of bounded agents who are able to write proofs about themselves andone another, including the capacity to out-perform classical Nash equilibriaand correlated equilibria, attaining mutually cooperative program equilibriumin the Prisoner's Dilemma. Previous cooperative program equilibria studied byTennenholtz (2004) and Fortnow (2009) have depended on tests for programequality, a fragile condition, whereas "L\"obian" cooperation is much morerobust and agnostic of the opponent's implementation.

Parametric Bounded L\"ob's Theorem and Robust Cooperation of Bounded Agents