This question's explanation and your answers so far.
A社と競合B社の相互の意思決定において，A社先手で相互に3手目までの打つ手を次の表のとおり作成した。表の数値は3手目の局面におけるA社の期待値である。A社は自社の期待値を大きくするように打つ手を選択し，B社は，3手目の結果を予測してA社の期待値を小さくするように打つ手を選択するとき，3手目の局面におけるA社の期待値は何億円か。 単位 億円
| 1手目：A社 | 2手目：B社 | 3手目：A社 | 3手目の局面におけるA社の期待値 |
|---|---|---|---|
| 値引きする | 値引きする | 広告する | 9 |
| 広告しない | 2 | ||
| 値引きしない | 広告する | 15 | |
| 広告しない | 8 | ||
| 値引きしない | 値引きする | 広告する | 0 |
| 広告しない | −6 | ||
| 値引きしない | 広告する | 10 | |
| 広告しない | 4 |
8 hundred million yen is the value of the cell for 1st move = cut price, 2nd move = do not cut price, 3rd move = do not advertise. However, the 3rd move on this branch is Company A's turn, and Company A chooses so as to make its own expected value larger, so it takes the 15 of Advertise on the same branch. 8 is an option that Company A itself throws away, so it never comes about in the first place. This is the mistake you fall into if you are conscious only of Company B minimizing and end up taking the smaller value at Company A's turn as well.
正解 The correct answer is イ (9 hundred million yen). This question expresses the situation in which the two sides choose moves alternately as a tree structure (a decision tree) and is solved by backward induction, fixing the value of each position by working back from the last move. Company A chooses the move that makes its expected value larger and Company B chooses the move that makes it smaller, so we take the maximum at positions where Company A moves and the minimum at positions where Company B moves, going back up toward the root. First, evaluate the 3rd move (Company A's turn, maximize). On the branch 1st move = cut price, 2nd move = cut price, the values are Advertise 9 and Do not advertise 2, so 9. On the branch 1st move = cut price, 2nd move = do not cut price, they are 15 and 8, so 15. On the branch 1st move = do not cut price, 2nd move = cut price, they are 0 and −6, so 0. On the branch 1st move = do not cut price, 2nd move = do not cut price, they are 10 and 4, so 10. Next, evaluate the 2nd move (Company B's turn, minimize). If Company A chose to cut price on the 1st move, Company B's options are 9 and 15, so Company B aims for the smaller one and chooses to cut price, and the value of this position is fixed at 9. If Company A chose not to cut price on the 1st move, Company B's options are 0 and 10, so Company B chooses to cut price, and the value of this position is fixed at 0. Finally, evaluating the 1st move (Company A's turn, maximize), Company A can choose between 9 and 0, so Company A chooses to cut price and obtains 9. Therefore, the expected value for Company A at the 3rd-move position that is realized when both sides choose their moves rationally is 9 hundred million yen, and the correct answer is イ.
10 hundred million yen is the value of the cell for 1st move = do not cut price, 2nd move = do not cut price, 3rd move = advertise. To go down this branch, Company A must first choose not to cut price on the 1st move, and Company B must then also choose not to cut price on the 2nd move. However, Company B chooses so as to make Company A's expected value smaller, so under 1st move = do not cut price it chooses to cut price, which gives 0, and the branch with 10 is pruned away. Even if Company A did choose not to cut price on the 1st move, the value it could obtain would be only 0, which is worse than cutting price, which gives 9, so Company A does not choose this branch either.
15 hundred million yen is the largest value in the table, but that cell is 1st move = cut price, 2nd move = do not cut price, which can be reached only on the premise that Company B cooperates. Company B moves so as to make Company A's expected value smaller, so under 1st move = cut price it does not choose do not cut price, which gives 15, but chooses cut price, which gives 9. Ignoring the existence of the opponent's turn and answering with the largest value in the table as it stands is the most common mistake in this type of question.