Rational Decision Theory I believe I am in The choice(s) that it thus and such a would be rationally Decision and Game Theory Decision and Game Theory circumstance. permissible for any x agent with these beliefs and these My preferences preferences to make. Justin C. Fisher regarding possible expected outcomes Southern Methodist University are as follows… Dept of Philosophy Game Theory Types of Game Theory Game theory is the study of the patterns that arise in situations Rational Game Theory: assumes each Rational Game Theory: where various agents each player chooses rationally (Phil, Econ) (rationally) make choices, where these choices may affect other Evolutionary Evolutionary Game Theory: Game Theory: studies which strategies would proliferate and agents. stay stable in competition with others (Bio, Econ, Phil) Human Game Theory: studies how actual Human Game Theory: people make choices in games (Psych, behavioral Econ) 1
Significance of our Decision Making in cases of Certainty ‘Irrationality’? 1. If I put my O up top, I will win. • There may be different notions of 2. If I don’t, I will lose. rationality. 3. I prefer winning to losing. • If game theory makes predictions 4. So I should put my O up top. based on the assumption that we’re rational in a way that we aren’t, then it will make bad predictions. If you’re certain that a particular choice will have the best available consequences, you should make that choice. Decision Making without Probabilities Warning: People reason strangely about risks 1. I don’t know how X will choose. Scenario 1: Get $1000 for sure. 2. If I put my O to the left, I might A. Get extra $500 for sure. Subjects are averse to win, lose, or tie. taking risks just for gains. B. 50% extra $1000, 50% gain 0 3. If I put my O in bottom middle, I can’t win but can force a tie. 4. I prefer win > tie > lose. Scenario 2: Get $2000 for sure. 5. So, …? C. Lose $500 of that for sure. Subjects are often willing to take risks D. 50% lose $500, 50% lose nothing. A “risk-tolerant” approach would accept to avoid losses. the risk in hopes of getting a win. “Maximin” maximizes the worst-case scenario: But these two scenarios are exactly the same!!! better to force a tie than risk a loss. (“risk averse”) Whatever’s rational in one must be rational in the other. Best in safe environment Best in a harsh environment that will Be wary when people frame things in terms of where much can be exploit any weakness. E.g., “zero sum” gains/losses – redescribe the other way too. gained and little lost. game vs smart well ‐ informed opponent. 2
Maximize Expected Value Computing the expected value of a choice 1. List the different possible ways the world might be. Suppose you can flip a coin. You’ll win $24 if it flips 2. Estimate how probable each possibility is. heads, but you’ll lose $36 if it flips tails. Should you? 3. Estimate how valuabe each possibility would be. 4. Multiply the probability of each possibility times its value. 5. Add up all those products That’s the expected value. 1/2 (chance you’ll lose) x -$36 Suppose you can flip a coin. You’ll win $24 if it flips + 1/2 (chance you’ll win ) x +$24 heads, but you’ll lose $36 if it flips tails. Should you? --------------------------------------------------- = -$6 “expected 1/2 (chance you’ll lose) x -$36 value” + 1/2 (chance you’ll win ) x +$24 You have prudential reason to perform whichever --------------------------------------------------- “expected available option has the highest expected value. value” = -$6 High Payoffs with Low Probability Probabilities and payoffs both matter. Suppose you can flip a coin. You’ll win $24 if it flips Suppose you can pay $1 to win $1,000,000 if the ace heads, but you’ll lose $36 if it flips tails. Should you? of hearts is randomly drawn from a deck of cards. The coin is weighted: it flips Heads 2/3 of the time. 51/52 (chance you’ll lose) x -$1 1/3 (chance you’ll lose) x -$36 + 1/52 (chance you’ll win ) x $1,000 + 2/3 (chance you’ll win ) x +$24 --------------------------------------------------- --------------------------------------------------- = $18 = +$4 “expected “expected value” value” If the payoff is small, the probability of winning has If the potential payoff is high enough, it can be a to be really high for it to be a good gamble. good gamble even if you’re very unlikely to win. 3
Prisoners’ Dilemma Causal Dominance Coope oopera rate te Coope oopera rate te with o ith other with o ith other priso risone ner Defec efect priso risone ner Defec efect 1 ye 1 year Free! ree! 1 year 1 ye Free! ree! Coope oopera rate te Coope oopera rate te with o ith other with o ith other priso risone ner priso risone ner 1 ye 1 year 5 y years ears 1 ye 1 year 5 y years ears 5 y years ears 3 y years ears 5 y years ears 3 y years ears Defec efect Defec efect Free! ree! 3 y years ears Free! ree! 3 y years ears Prisoners’ Dilemma But… • If only they had both cooperated, Coope oopera rate te they both would have been better with o ith other off! priso risone ner Defec efect 1 ye 1 year Free! ree! Coope oopera rate te • Who’s more rational – the with o ith other defectors who are thrown away for priso risone ner 1 ye 1 year 5 y years ears years, or the cooperators who get off lightly? 5 y years ears 3 y years ears Defec efect Free! ree! 3 y years ears 4
A trick… Rational Prisoners’ Dilemma • In framing the last comparison, I • Suppose that, in addition to assumed that both prisoners will knowing all the standard stuff, end up choosing the same. the prisoners both know that they are both rational. • This assumption isn’t justified in the simplest description of • Does this knowledge change what the problem. choices they ought to make (or what choices norms of rationality • But this assumption would be will lead them to make)? justified if you knew both prisoners would decide in roughly the same ways. Rational PD Psychologically Similar PD • Suppose that, in addition to knowing all the standard stuff, Coope oopera rate te with o ith other the prisoners both know that they priso risone ner Defec efect have been paired with each other because they are psychologically 1 ye 1 year Free! ree! Coope oopera rate te similar. with o ith other priso risone ner 1 ye 1 year 5 y years ears • Does this knowledge change what 5 y years ears 3 y years ears choices they ought to make (or Defec efect what choices norms of rationality Free! ree! 3 y years ears will lead them to make)? 5
Hofstadter’s Argument How many iterations? • We’re both rational. Single-shot – the full impact of Single-shot • Whatever norms of rationality dictate your choice is captured in the to me, they’ll also dictate to you. matrix above. No one will • So we’ll both choose the same thing. remember what you did and further • If we both choose C we’ll each get -1. reward or punish you for it. • If we both choose D we’ll each get -3. • Rational people would get -1 rather Iterated Iterated – there’s a good chance than -3. you’ll be paired off against this -------------------------------------- partner again, and what you do • So we’ll rationally choose C. this time may affect how he/she treats you later on. Axelrod’s Keys to Success in Some Strategies for Iterated PD’s Iterated PD’s All-C ll-C – – Always cooperate. Nicen icenes ess – s – Start by cooperating. Provo rovoca cabi bili lity - don’t keep cooperating All-D ll-D – Always defect. with someone who will abuse you. Tit-f it-for or-T -Tat at – – Always cooperate, except Forgi orgivi ving ngne ness – – If the other player defect once in response to defection. will repent for a mistake, you’re better off going back to mutual cooperation with them. Tit-f it-for or-T -Two wo-Tats – – Always cooperate, except defect once in response to Clari larity ty – – Don’t try to be too fancy, or being defected against twice in a row. you’ll make other players suspicious and lose the chance for cooperation. 6
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