1. Uncertainty and Games (Ch. 15)

2. Uncertainty If a game outcome is certain can it achieve meaningful play? –Example of such a game? Two kinds of uncertainty: –Macro-level (overall game) –Micro-level (individual player’s actions)

3. Uncertainty and Categories of Games According to AI Perfect information Imperfect information Deterministic Chance Yes For which of the following categories, label with “Yes” those for which games can have uncertainty Lesson: you don’t have to “rolling a dice”, to achieve uncertainty in a game

4. Feeling of Randomness We can even achieve a feeling of randomness in a deterministic perfect information game –Example? Danger: designing a chaotic game –Example?

5. Probability in Games Probability: a mathematical formalization of uncertainty Examples of games using probability: In a game like Chutes and Ladders, player is not making decisions (probabilities – a random number generator is deciding-), why is it “fun”?  chance to hit  amount of damage dealt  Next shape in Tetris  Initial location in a multiplayer RTS game  Random loot in an MMO  …

6. Probability Example. Suppose that you are in a TV show and you have already earned 1’000.000 so far. Now, the host propose you a gamble: he will flip a coin if the coin comes up heads you will earn 3’000.000. But if it comes up tails you will loose the 1’000.000. What do you decide? We know a degree of belief Probability theory allows the analysis of decisions based on the degree of belief

7. Probability Suppose that I flip a “fair” coin:  what is the probability that it will come heads: 0.5 Suppose that I flip a “totally unfair” coin (always come heads):  what is the probability that it will come heads: 1 Assigns a number between 0 and 1 to events The closer an event is to 1, the more likely we believe it will occur The closer an event is to 0, the less likely we believe it will occur

8. Another Example Example: 1000 tickets are sold at a value of $1 each 100 are selected. The first 90 get a $5 price, the next 9 get a $10 price and the last will get $100 Probability of having a winning ticket:100/1000 = 0.1 Probability of holding a $5 ticket: 90/1000 = Probability of holding a $10 ticket: 9/1000 = Probability of holding the $100 ticket: 1/1000 = Probability of holding a losing ticket: 900/1000 =.09.009.001.9

9. Probability Distribution The events E 1, E 2, …, E k must meet the following conditions: One always occur No two can occur at the same time The probabilities p 1, …, p k are numbers associated with these events, such that 0  p i  1 and p 1 + … + p k = 1 A probability distribution assigns probabilities to events such that the two properties above holds

10. Example (Probability Distribution) In the example: E 1 = “holding a $5 ticket” E 2 = “holding a $10 ticket” E 3 = “holding a $100 ticket” E 4 = “holding a losing ticket” The probabilities are: p 1 =.09 p 2 =.009 p 3 =.001 p 4 =.9 Note that the probabilities add to 1 Could we add E 5 = “holding a winning ticket”? No! because E 5 occurs at the same time as E 1, E 2 and E 3

11. Expected Utility Coming back to the example: Answer to that depends on: – the probability of winning $0 or $ 3’000.000 –How much money you currently have –Expected utility: a measure of how much I win from taking an action Suppose that you are in a TV show and you have already earned 1’000.000 so far. Now, the host propose you a gamble: he will flip a coin if the coin comes up heads you will earn 3’000.000. But if it comes up tails you will loose the 1’000.000. What do you decide?

12. Warnings about Probabilities (1) A random function assigns to k events: E 1, E 2, …, E k, the same probability: 1/k (called “uniform distribution”)  Example: rolling a dice has 6 events (one for each face) with probability of each occurring been 1/6 Problem: Computers cannot make a perfectly random function  It is “random” in the sense that you cannot predict the outcome in advance  But it is not a uniform distribution  This is due to the fact that computers are intrinsically deterministic

13. Warnings about Probabilities (2) People do not always follow the rules of probability:  Experiment with people  Choice was given between A and B and then between C and D: A: 80% chance of $4000 B: 100% chance of $3000 C: 20% chance of $4000 D: 25% chance of $3000 Majority choose B over A and C over D  This turns out to be mathematically inconsistent with the expected utility Book discusses other miss-conceptions

14. Final Thought Probability does not equate to chance in games But well laid-out element of chance will result in meaningful choices Example: Player needs to decide whether to attack a monster or not based  This decision is based on expected utility  A “feeling” of how success will it be  Sometimes players can get quite formalformal  How worth would it be if successful