978-0393919684 Chapter 6 Solution Manual Part 4

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U10. (a) The game table for the first-stage game follows:

Nancy

Yes No

Monica

(b) The two pure-strategy Nash equilibria are indicated by double borders in the preceding

U11. (a) The profit functions when both firms decide to invest are

Solving the first-order conditions, the best-response functions of the two firms are

The Nash equilibrium occurs at the values of QB and QC that satisfy both best-response functions

simultaneously:

The equilibrium price and profits when both firms invest are

(b) The profit functions when both firms decide not to invest are

Solving the first-order conditions, the best-response functions of the two firms are

Solving the best-response functions simultaneously to find the Nash equilibrium quantities:

The equilibrium price and profits when neither firm invests are

Solving the first-order conditions, the best-response functions of the two firms are

Solving the best-response functions simultaneously, the Nash equilibrium quantities are

The equilibrium price and profits when Bilge invests and Chem does not are

(d) The game table for the first-stage investment decision is shown below. Since the firms

face the same costs depending on their investment decision, the payoffs for the cell (No investment,

Investment) are simply the reverse of the payoffs for (Investment, No investment).

Chem

Investment No investment

(e) In the first-stage game, Investment is the dominant strategy for each firm. Equilibrium

U12. As usual for sequential-move games, we begin at the end. Suppose both aristocrats have walked 5

steps and are right next to each other, but neither has yet fired. The payoffs are shown next, and it is easy

to see that Shoot is the dominant strategy at this step for each player; the resulting payoffs are (0, 0), as

shown below:

Chagrin

Shoot Not

Knowing the equilibrium at the 5-step subgame, we use rollback to consider the payoffs after each player

has walked 4 steps, as shown below:

Chagrin

Shoot Not

The preceding payoffs are derived as follows:

(i) Top right cell: If Renard shoots and Chagrin does not, then Renard hits with probability 0.8

and gets 1, and misses with probability 0.2, in which case Chagrin has a sure shot after 5 moves, so

(iii) Bottom right cell: If neither shoots on step 4, the game goes to step 5, where we know the

For step 3, the duelists are 6 steps apart and the probability of hitting if you shoot is 0.6. Similar

Chagrin

Shoot Not

Again, Shoot is dominant for each player, yielding an outcome with payoffs of (0, 0).

Next move back to step 2, with the pair 8 paces apart and the probability of hitting down to 0.4.

The payoff table follows:

Chagrin

Shoot Not

Things have changed: Not shoot is now the dominant strategy for both. The payoffs from the resulting

outcome are still (0, 0).

Finally, at step 1, with the pair in their starting positions and the probability of hitting only 0.2,

the payoff matrix follows:

Chagrin

Shoot Not

Not shoot is again the dominant strategy, again yielding an outcome with payoffs of (0, 0).

Actual play of the game will result in both players shooting on step 3. The full equilibrium

strategy for either player is “Do not shoot on steps 1 and 2 no matter what. At any step, if the other player

U13. There are potentially many examples of business competition that parallels the aristocrats’ duel.

One such example is the race to get new software or gadgets to the market first. In this context there is a