Worst Case vs Average Case

Stochastic Games

• Why would we not know what the result of an action will be?
• Explicit randomness: rolling dice
• Unpredictable opponents: the ghosts respond randomly
• Actions can fail: when moving a robot, wheels might slip
• Values should now reflect average-case (expectimax) outcomes, not worst-case (minimax) outcomes

Expectimax Search

• Expectimax search: compute the average score under optimal play
• Max nodes as in minimax search
• Chance nodes are like min nodes but the outcome is uncertain
• Calculate their expected utilities
• i.e., take weighted average (expectation) of children
• Later, we will learn how to formalize the underlying uncertain-result problems as Markov Decision Processes

Expectimax Implementation

Expectimax Implementation

Expectimax Example

Expectimax Pruning

Depth-Limited Expectimax

Primer: Probabilities

• A random variable represents an event whose outcome is unknown.
• A probability distribution is an assignment of weights to outcomes
• Example: Traffic on freeway
• Random variable: $T = \mbox{ whether there is traffic}$
• Outcomes: $T \in \{none, mild, heavy\}$
• Distribution: $P(T=none) = 0.25$, $P(T=mild) = 0.50$, $P(T=heavy) = 0.25$

Primer: Probabilities...

• Some laws of probability (more later):
• Probabilities are always non-negative
• Probabilities over all possible outcomes sum to one
• As we get more evidence, probabilities may change:
• $P(T=heavy) = 0.25$, $P(T=heavy \mid Hour=8am) = 0.60$
• We will talk about methods for reasoning and updating probabilities later

Primer: Expectations

• The expected value of a function of a random variable is the average, weighted by the probability distribution over outcomes
• Example: How long to get to the airport?

What probabilities to use?

• In expectimax search, we have a probabilistic model of how the opponent (or environment) will behave in any state
• Model could be a simple uniform distribution (roll a die)
• Model could be sophisticated and require a great deal of computation
• We have a chance node for any outcome out of our control: opponent or environment
• The model might say that adversarial actions are likely.
• For now, assume each chance node magically comes along with probabilities that specify the distribution over its outcomes

Quiz: Informed Probabilities

• Let us say you know that your opponent is actually running a depth 2 minimax, using the result 80% of the time, and moving randomly otherwise.
• Question: What tree search should you use?
• Answer: Expectimax
• To figure out EACH chance node’s probabilities, you have to run a simulation of your opponent
• This kind of thing gets very slow very quickly
• Even worse if you have to simulate your opponent simulating you
• Except for minimax, which has the nice property that it all collapses into one game tree

The Dangers of Optimism and Pessimism

Assumptions vs Reality

Mixed Agent Games

Mixed Agent Games: Backgammon

Maximum Expected Utility

• Why should we average utilities? Why not minimax?
• Principle of maximum expected utility:
• A rational agent should chose the action that maximizes its expected utility, given its knowledge
• Questions:
• Where do utilities come from?
• How do we know such utilities even exist?
• How do we know that averaging even makes sense?
• What if our behavior (preferences) cannot be described by utilities?

What utilities to use?

Utilities

• Utilities are functions from outcomes (states of the world) to real numbers that describe an agent's preferences
• Where do utilities come from?
• In a game, may be simple $(+1/-1)$
• Utilities summarize the agent's goals
• Theorem: any "rational" preferences can be summarized as a utility function
• We hard-wire utilities and let behaviors emerge
• Why don't we let agents pick utilities?
• Why don't we prescribe behaviors?