Decision Theory Basics Decision theory A decision involves
Decision Theory: Basics
Decision theory A decision involves a choice between two or more acts, each of which will produce one of (possibly) several outcomes. Divide decisions into three categories: 1. Decisions under certainty 2. Decisions under risk 3. Decisions under uncertainty (a. k. a. Decisions under ignorance)
Decisions Under Certainty The agent is certain what outcome each act will produce. They are the easiest decisions to make: Choose the act that will produce the outcome you most prefer!
Example If you raise your right hand, I will give you $100. If you raise your left hand, I will give you $10. If you raise neither hand, I will not give you anything.
Example If you raise your right hand, I will give you $100. If you raise your left hand, I will give you $10. If you raise neither hand, I will not give you anything. Right Left Neither Both $100 $10 $0 $110
Decisions Under Risk These are decisions where : 1. ) You lack certainty about the what outcomes the actions will have. 2. ) But you can at least assign probabilities to what outcomes the actions will have.
Example You have two options. Option A: I’ll flip a coin. If it lands heads, I’ll give you $50. If it lands tails, I won’t give you anything. Option B: I’ll pick a number between 21 -30 inclusive. If you guess the number, I’ll give you $70. If you don’t guess number, I won’t give you anything.
Which option do you choose? 1. Option A 2. Option B
Decisions Under Ignorance These are decisions where : 1. ) You lack certainty about the what outcomes the actions will have. 2. ) You can’t even assign probabilities to the outcomes the actions will have.
Example Suppose that just after graduating college you are offered three jobs. First, the Exe Company offers you a salary of $40, 000. Exe is well established and secure. Your next offer comes from the Wye Company. Here, the salary is $60, 000. But Wye is a new company and is less secure. You do not know how likely it is to last. It might go bankrupt and then you’ll be left without a job. The final offer comes from the Zee Company, which is as well established and secure as Exe. Here, the salary is $65, 000. In all other respects, the jobs are similar. Which job do you take?
Wye does not go bankrupt Wye goes bankrupt Take job at Exe $40, 000 Take job at Wye $60, 000 $0 Take job at Zee $65, 000
Wye does not go bankrupt Wye goes bankrupt Take job at Exe $40, 000 Take job at Wye $60, 000 $0 Take job at Zee $65, 000 Taking the job at Zee dominates the other acts.
Wye does not go bankrupt Wye goes bankrupt Take job at Exe $40, 000 Take job at Wye $60, 000 $0 Now what?
Sonny and Cher
Sonny is “The Gambler” Wye does not go bankrupt Wye goes bankrupt Take job at Exe $40, 000 max = $40, 000 Take job at Wye $60, 000 $0 max= $60, 000 Chooses the best of the best. Maximax Strategy
Cher is “Cautious” Wye does not go bankrupt Wye goes bankrupt Take job at Exe $40, 000 min = $40, 000 Take job at Wye $60, 000 $0 min = $0 Chooses the best of the worst. Maximin strategy
Who is right?
Maximax: The Gambler This strategy tells us to maximize the maximum. To apply maximax we first need to find the maximum outcome for each act. Simply read across a row and at the end of the row write down the maximum value; then choose the act with the maximum of the maximum values. S 1 A 1 + $4 A 2 + $2 S 2 - $1, 000 max = + $4 - $1 max = + $2 According to maximax, you should choose A 1.
Which action does maximax pick? S 1 A 1 + $4 A 2 + $2 1. A 1 2. A 2 S 2 - $1, 000 max = + $4 - $1 max = + $2
Criticism of Maximax Too risky! Prohibits us from avoiding huge losses where there are only slight gains. S 1 A 1 + $4 A 2 + $2 S 2 - $1, 000 max = + $4 - $1 max = + $2
Which action does maximin pick? S 1 A 1 + $4 A 2 + $2 1. A 1 2. A 2 S 2 - $1, 000 max = + $4 - $1 max = + $2
Maximin: The Cautious Player This strategy tells us to maximize the minimum. To find the minimum for each act, read across the row and write down the minimum value; then choose the act that has highest of lowest values S 1 A 1 + $4 A 2 + $2 S 2 - $1, 000 min = -$1, 000 - $1 min = - $1 According to maximin, you should choose A 2.
Criticism of Maximin Too conservative. Prohibits us from taking advantage of huge gains where there are only slight losses. S 1 S 2 A 1 $1. 50 $1. 75 A 2 $1. 00 min = $1. 50 $10, 000 min = $1. 00
Principle of Insufficient Reason When faced with a decision under ignorance, treat all outcomes as equiprobable. In effect, you chose in terms of expected utility. Take job at Exe Take job at Wye Exe: Wye does not go bankrupt Wye goes bankrupt $40, 000. 5 $60, 000 . 5 $0 . 5 ($40 k x. 5) + ($40 k x. 5) = $40, 000 ($60 k x. 5) + ($00 k x. 5) = $30, 000 . 5
Criticism of the Principle of Insufficient Reason ARBITRARY! Gives no reason for preferring one probability distribution over another. Take job at Exe Take job at Wye does not go bankrupt Wye goes bankrupt $40, 000. 5 $60, 000 . 5 $0 . 5
Decision Strategies Principle of Insufficient Reason: Assume every state is equally probable and choose the act that, on this assumption, maximizes expected utility. Dominance: If, no matter what happens, an act is never worse and sometimes better than every other, then choose it. Maximax: Choose the best of the best. Maximin: Choose the best of the worst. Minimax Regret: Choose the act that minimizes the maximum amount of regret.
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