�✁ ❙ ❜ ❘ ❩ ❣ ❫ ❘ ❛ ❙ ❩ ❣ ❫ ❛ ❫ ❘ ❩ ❢ ✌◗ ✞ ✁ ✌ ✄ ✌ P ✌ ❙ ❩ ❡ ❛ ❱ ❲ ❥ ❢ ✐ ❛ ❙ ❛ ❘ ❫ ✐ ❘ ❘ ❩ ❢ ❫ ✐ ❛ ❙ ❩ ❤ ❫ ❙ ❛ ✄ ☞ ❘ ✺ ✴ ✶ ✱ ✻ ✻ ✳ ✲ ✲ ❁ ❀ ✿ ✾ ✿❃ ✽ ✲ ✳ ✴ ✻✼ ✺ ✷ ✸✹ ✶✷ ✂ ✱✲✳✴ ❂ ✵ ☛ ❄ ✁ ☎ ✟ ✂ ☛ ❞ ❝ ▲ ❝ ❏ ❈ ✳ ❄ ■ ❍ ✱ ● ❋ ❉❊ ❉ ❈ ❇ ❆ ✼ ❅ ❲❳ ❨ ❘ ✺ ✲ ❁ ❀ ✿ ✺ ✾ ✽ ✲ ✳ ✴ ✻✼ ✷ ✳ ✸✹ ✶✷ ✵ ✱✲✳✴ ✰ ❫ ❴ ❙ ❳ ❧ ❪ ✲ ✻ ❨ ❉❊ ▲ ♥ ❏ ❈ ❄ ✳ ■ ❍ ✱ ● ❋ ❉ ✻ ❈ ❇ ❆ ✼ ❅ ❄ ✵ ✿❃ ❂ ✴ ✶ ✱ ❬ ❘ ❬ ❘ ❙ ❲❳ ❱ ❜ ❴ ✐ ❳ ❲ ❪ ❬ ❨ ❲❳ ❬ ❱ ❢ ❙ ❜ ❘ ❙ ❜ ❴ ✐ ❲❳ ❪ ❨ ❪ ❳ ❘ ❧ ❱ ❜ ❛ ❴ ❙ ❳ ❲ ❪ ❬ ❨ ❲❳ ❲❳ ❱ ♠ ❧ ❦ ❲ ❩ ❢ ❙ ❛ ❘ ❴ ✐ ✰ ✵ ♥ ❁ ✵ ✿❃ ❂ ✴ ✶ ✱ ✻ ✻ ✳ ✲ ✲ ❀ ❅ ✿ ✺ ✾ ✽ ✲ ✳ ✴ ✻✼ ✺ ✷ ✸✹ ✶✷ ❄ ✼ ✱✲✳✴ ❑ ◆ ◆ ✌ ✁ ✟ ☞ ✂ ✝ ▼ ❑ ▲ ❏ ❆ ❈ ❄ ✳ ■ ❍ ✱ ● ❋ ❉❊ ❉ ❈ ❇ ✵ ✰ ◆ ✞ ✁ ✌ ✍ ✟ ☞ ☞ ✂✌ �✁ ☞ ☛ ✟ ✟✡ ✌ ✂ ✄ ✠ ☎ ✂ ✁ ☎ ✟ ✂ ✝✞ ☎✆ ✄ ✞ ✎ ✮✯ ✙ ✭ ✪✬ ✩✪✫ ★ ✧ ✥✦ ✣✤ ✢ ✜ ✛ ✚ ✔ ✏ ✓ ✒ ✙ ✒ ✘ ✖✗ ✕ ✏ ✔ ✓ ✒ ✑ ❙ ◆ ❈ ❄ ✌ ✄ ❖ ❈ ▲ ❈ ❏ ❙ ✳ ✌ ■ ❍ ✱ ● ❋ ❘ ❉❊ ❙ ❵ P ✄ ❉ ❲❳ ❴ ❙ ❳ ❫ ❲ ❪ ❨ ❘ ❱ ✌ ❯ ❚ ❘ ❙ ❛ ❘ ✌◗ ✞ ✁ ❜ ❘ ❈ ✶✷ ✲ ✳ ✴ ✻✼ ✺ ✷ ✸✹ ❘ ✵ ✾ ✱✲✳✴ ❙ ✰ ❘ ◆ ❛ ❜ ❙ ◆ ❇ ✽ ✺ ✿ ❆ ✼ ❅ ❄ ✵ ❙ ✿❃ ❂ ✴ ✶ ✱ ✻ ✻ ✳ ✲ ✲ ❁ ❀ Notation: uncertain prizes An agent chooses among prizes ( Lottery ❩❭❬ indifference between preferred to not preferred to , , etc.) and lotteries , i.e., situations with and L 1−p p B A Constraints: Rational preferences Idea: preferences of a rational agent must obey constraints. Value of information Decision networks Multiattribute utilities Money Utilities Rational preferences Monotonicity Substitutability Continuity Transitivity Orderability behavior describable as maximization of expected utility
❞ ✰ ✳ ✴ ✻✼ ✺ ✷ ✸✹ ✶✷ ✵ ✱✲✳✴ ⑨ ✽ ❚ ❜ ❘ ❲ ❫ ❲ ❪ ❬ ❩ ✲ ✾ ⑩ ✴ ❈ ❇ ❆ ✼ ❅ ❄ ✵ ✿❃ ❂ ✶ ✺ ✱ ✻ ✻ ✳ ✲ ✲ ❁ ❀ ✿ ❷ ❲ ❉❊ ✴ ❈ ❇ ❆ ✼ ❅ ❄ ☛ ✿❃ ❂ ✶ ❉❊ ✱ ✻ ✻ ✳ ✲ ✲ ❁ ❀ ✿ ❉ ❋ ❶ ✟ ⑩ ⑨ ❚ ❘ ✌◗ ✟ ✂ ✟ ☞ ✂ ● ⑧ ⑦ ▲ ❏⑦ ❈ ❄ ✳ ■ ❍ ✱ ❉ ❋ ✾ ✾ ✻ ✻ ✳ ✲ ✲ ❁ ❀ ✿ ✺ ✽ ✶ ✲ ✳ ✴ ✻✼ ✺ ✷ ✸✹ ✶✷ ✵ ✱ ✴ ✰ ❋ ▲ ❏➁ ❈ ❄ ✳ ■ ❍ ✱ ● ❉❊ ❂ ❉ ❈ ❇ ❆ ✼ ❅ ❄ ✵ ✿❃ ✱✲✳✴ ❸ ● ⑧ ☛ ✞ ◗ ✈ ✂ ✟ ☞ ✟ ✂ ❏ ✌◗ ▲ ❏ ❏ ❈ ❄ ✳ ■ ❍ ✱ ☞ ⑩ ➀ ❯ ① ❽ ❿ ❽ ❾ ❫ ✇ ① ❽ ❫ ❶ ❸ ③ ❸ ❯ ❷ ⑩ ❸ ③ ❬ ❯ ✺ ✵ ✽ ✻ ✼ ❅ ❄ ✵ ✿❃ ❂ ✴ ✶ ✱ ✻ ❇ ✳ ✲ ✲ ❁ ❀ ✿ ✺ ✾ ✽ ❆ ❈ ✳ q ✁ ✟ ✟✉ ✟t s ☛ r q ▲ ❏ ❉ ❈ ❄ ✲ ■ ❍ ✱ ● ❋ ❉❊ ✲ ✴ ✌ ✌ ✂ ✁ ☎ ✞ ✌◗ ✞ ✁ ✌ ✄ P ♦ ✌ ✄ ❡ ☞ ☛ ✁ ☎ ✟ ✂ ✆ ❙ ✻✼ ❘ ✺ ✷ ✸✹ ✶✷ ✵ ✱✲✳✴ ✰ ♣ ✐ ❘ ❛ ❛ ✐ ❘ ❙ ❙ ❛ ❘ ❙ ✐ ✐ ✍ ✳ s ③ ④ ② ❳ ④ ❲ ❨ ③ ③ ❫ ❨ ① ② ❳ ❡ ❲ ❱ ❩ ❴ ❯ ❙ ✱✲✳✴ ✳ ✴ ✻✼ ✺ ✷ ✸✹ ✶✷ ✵ ✰ ⑤ ❫ ⑥ ② ❩ ✇ ⑥ ❲ ⑥ ✇ ① ➁ ✂ ❩ ✇ ❦ ❘ ❩ ✇ ✇ ✈ ✟ ❫ ☞ ✟ ✂ ✝ ✆ ✂✌ ✞ ✌ ❙ ❫ ♠ ❜ ❛ ❘ Violating the constraints leads to self-evident irrationality Theorem (Ramsey, 1931; von Neumann and Morgenstern, 1944): Given preferences satisfying the constraints there exists a real-valued function such that For example: an agent with intransitive preferences can be induced to give away all its money A If , then an agent who has would pay (say) 1 cent to get MEU principle : 1c 1c Choose the action that maximizes expected utility If , then an agent who has would pay (say) 1 cent to get Note: an agent can be entirely rational (consistent with MEU) B C without ever representing or manipulating utilities and probabilities If , then an agent who has 1c would pay (say) 1 cent to get E.g., a lookup table for perfect tictactoe Normalized utilities : Utilities map states to real numbers. Which numbers? , Standard approach to assessment of human utilities: Micromorts : one-millionth chance of death compare a given state to a standard lottery that has useful for Russian roulette, paying to reduce product risks, etc. “best possible prize” with probability “worst possible catastrophe” with probability QALYs : quality-adjusted life years adjust lottery probability until useful for medical decisions involving substantial risk Note: behavior is invariant w.r.t. linear transformation continue as before 0.999999 where ✇❺❹ pay $30 ❩❼❻ ❩❼❻ ~ L With deterministic prizes only (no lottery choices), only 0.000001 instant death ordinal utility can be determined, i.e., total order on prizes
➂ ❄ ● ❋ ❉❊ ❉ ❈ ❇ ❆ ✼ ❅ ✵ ❍ ✿❃ ❂ ✴ ✶ ✱ ✻ ✻ ✳ ✲ ✱ ■ ❁ ☞ ✝ ➋ ✟ ✄ ✂ ✂ ☛ ✟ ✂ ✝ ✳ r ❑ ❈ ▲ ❑ ❈ ❏ ❈ ❄ ✲ ❀ ✝ ✳ ✞ ✌ ➈ ❉ ▲ ❉ ❏ ❈ ✂ ■ ◗ ❍ ✱ ● ❋ ❉❊ ❉ ❈ ❇ ❆ ✟ ✟ ✿ ✸✹ ✺ ✾ ✽ ✲ ✳ ✴ ✻✼ ✺ ✷ ✶✷ ☎ ✵ ✱✲✳✴ ✰ ◗ ✄➊ ☎ ✂➉ ✌ ✁ ✁ ✂✌ ✂ ❅ ✿ ✶ ✱ ✻ ✻ ✳ ✲ ✲ ❁ ❀ ✺ ❂ ✾ ✽ ✲ ✳ ✴ ✻✼ ✺ ✷ ✸✹ ✴ ✿❃ ✵ ❍ ❈ ▲ ❈ ❈ ❏ ❈ ❄ ✳ ■ ✱ ✵ ● ❋ ❉❊ ❉ ❈ ❇ ❆ ✼ ❅ ❄ ✶✷ ✱✲✳✴ ✟ ④ ❳ ➎ ➒ ➔→ ➓ ❳ ➒ ➎➏➐➑ ✇ ➌ ➔ ③ ③ ③ ① ➌ ✈ ✂ ✟ ☞ ✐ ➒ ✰ ✇ ❫ ④ ❻ ❳ ③ ③ ③ ❳ ① ❫ ➐ ④ ❻ ❳ ③ ③ ③ ❳ ① ✇ ❫ ✼ ❄ ❄ ❉ ❄ ✳ ■ ❍ ✱ ● ❋ ❉❊ ❈ ❏ ❇ ❆ ✼ ❅ ❄ ✵ ✿❃ ❂ ❈ ❊ ✶ ❩ ✇ ➆ ❫ ❚ ❩ ✇ ❫ ❚ ➅ ▲ ➃➄ ❚ ✈ ✌ ✁ ✵ r ❊ ✴ ✱ ➃➄ ❡ ❻ ✈ ✂ ✟ ☞ ✟ ✂ ✝ ✝ ✰ ☎ ✄ ✍ ✂ ✁ ✌ ✆ ✝ ❲ ✱✲✳✴ ✻ ✾ ✻ ✳ ✲ ✲ ❁ ❀ ✿ ✺ ✽ ✵ ✲ ✳ ✴ ✻✼ ✺ ✷ ✸✹ ✶✷ ❩ ☎ ➅ ✾ ✰ ✱✲✳✴ ✵ ✶✷ ✸✹ ✷ ✺ ✻✼ ✴ ✳ ✲ ✽ ✺ ➄ ✿ ❀ ❁ ✲ ✲ ✳ ✻ ✻ ✱ ✶ ✴ ❂ ✿❃ ❩ ❈ ❴ ➇ ❫ ❲ ❪ ❩ ❨ ➄ ❲❳ ➇ ❱ ❻ ❲ ❫ ❫ ❚ ❳ ❬ ❸ Money does not behave as a utility function For each , adjust until half the class votes for lottery (M=10,000) Given a lottery with expected monetary value , usually , i.e., people are risk-averse p Utility curve: for what probability am I indifferent between a fixed prize and 1.0 a lottery for large ? 0.9 0.8 Typical empirical data, extrapolated with risk-prone behavior: 0.7 +U 0.6 o o o o o o o o 0.5 o o o o +$ 0.4 −150,000 800,000 0.3 o o 0.2 0.1 0.0 o $x 0 500 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Add action nodes and utility nodes to belief networks to enable rational decision making How can we handle utility functions of many variables ? E.g., what is ? ❩❭➍ Airport Site How can complex utility functions be assessed from preference behaviour? Deaths Air Traffic Idea 1: identify conditions under which decisions can be made without com- plete identification of Litigation Noise U ❩❼❻ Idea 2: identify various types of independence in preferences and derive consequent canonical forms for Construction Cost ❩❼❻ Algorithm: For each value of action node compute expected value of utility node given action, evidence Return MEU action
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