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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
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(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
(c) The profit functions when Bilge decides to invest while Chem does not are
Solving the first-order conditions, the best-response functions of the two firms are
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
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
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
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
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
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
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
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
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
