How to Draw a Decision Tree 1. Put probabilities on the branches that are moves by Nature. 2. Put payoffs at the end points. Do not put expected payoffs there. If the player reaches a particular end point, he gets the actual payoff from that outcome, not the expected payoff. Expected payoffs are part of the players’ calculations in the middle of the game, not part of the description of the game. 3. The difference when we move to game theory is that there is more than one player, and each end point has to list a payoff for each player. 4. In both decision theory and game theory, look ahead and reason back. Cross out (mentally, at least) branches that are not going to be chosen.
Example Your company is thinking about investing 100 million dollars in California. You know there is a 70 percent chance of losing the entire amount, and a 30 percent chance of coming out with a net profit of 200 million dollars. You have the opportunity of first investing 20 million in Oregon, however. There, your chance of losing the 20 million is .5 and your chance of coming out 10 million ahead is also .5. Oregon provides information. If you succeed in Oregon, there is a 40 percent chance you will succeed in California. If you fail in Oregon, there is a 30 percent chance you will succeed in California. What do you do?
The Decision Tree Work back from the end. If you don’t invest in Oregon, the expected value of investment in California is .3(200)+.7(-100) = 60-70 = -10, so you shouldn’t invest. If you invested in Oregon and were successful, the expected value of investment in California would be (.4) 210+.6(-90) = 88-54 = 34, better than the 10 you would get by not investing. So do invest. If you invested in Oregon and were unsuccessful, the expected value of investment in California would be (.2) 180+.8(-120) =36-96 = -60, worse than the -20 you would get by not investing. So don’t invest. Now go to the first decision about whether to invest in Oregon. Not investing yields 0. Investing has an expected value of .5(34) + .5 (-20)= 17 -10 = 7. So you should invest in Oregon (and then invest in California only if the Oregon investment is successful).
The May Ball Casino Whoever has the most chips gets a free ticket for next year’s ball. Barry has $700 and Susan has $300 and there is time for one more spin of the wheel. Susan offers to split the value of the prize. Should Barry accept? Someone who bets on Red or Black has an 18/37 probability of winning double his bet. Someone who bets on the number being a multiple of 3 has a 12/37 probability of winning triple his bet. Susan bet $300 on the multiple of 3. Barry bet $200 on Black. Was that his best response? (see p. 29, DN book)
THE BID-ASK SPREAD Suppose someone is trading common stock shares in Fastcleaners, Inc. He and the rest of the market, think the value of the stock is $41/share. Yet: He bids $40/share to buy. He asks $42/share to sell, He wouldn’t buy for $40.50, or sell for $41.50. Why not? Why is there a spread?