# Knowledge, probability, and adversaries (preliminary report)

## Abstract

What should it mean for an agent to know or believe an assertion is true with probability 0.99? Different papers give different answers, choosing to use quite different probability spaces when computing the probability an agent assigns to an event. We show that each choice can be understood in terms of a betting game, and that each choice corresponds to betting against a different opponent. We consider three types of adversaries. The first selects the outcome of all nondeterministic choices in the system; the second represents the knowledge of the agent's opponent (this is the key place the papers mentioned above differ); the third is needed in asynchronous systems to choose the time the bet is placed. We illustrate the need for considering all three types of adversaries with a number of examples. Given a class of adversaries, we show how to assign probability spaces to agents in a way most appropriate for that class, where 'most appropriate' is made precise in terms this betting game. We conclude by showing how different assignments of probability spaces (corresponding to different opponents) yield different levels of guarantees in coordinated attack.