Venture Capital and the Finance of Innovation, Second Edition

Problems
(a) Draw the extensive form for this game.
(b) Draw the normal form for this game and solve for all Nash equilibria.
Solutions
(a) The extensive form is given in Exhibit 23-9. From this extensive form, one might think
that more graphics is a “better” strategy because it gives higher payoffs when the players
choose different strategies. This kind of reasoning is dangerous, because, as we will see, a
pure strategy of more graphics is not part of any NE.
EXHIBIT 23-9
LEADER-FOLLOWER GAME, EXTENSIVE FORM
Leadco
1
More
graphics
Faster
speed
2
Followco
3
23.2 SIMULTANEOUS GAMES 429
More
graphics
Faster
speed
More
graphics
Faster
speed
($6B, $2B)
($5B, $4B)
($4B, $5B)
($6B, $2B)
(b) The normal form for this game, with best responses circled, is given in Exhibit 23-10.
As in the odds-and-evens game, we find no pure-strategy NE. The reason is that
Leadco always wants to be the same as Followco, whereas Followco wants to be different.
With a simultaneous game, the best strategy is to try to keep the other company guessing.
The game-theoretic way to do this is with a mixed strategy.
To find the mixed-strategy equilibrium, we follow the same steps as we did for the
odds-and-evens game. Let p be the probability of Leadco playing more graphics, so that 1 p
is the probability of Leadco playing faster speed. With these probabilities, if Followco plays
more graphics then it would receive an expected payoff of
p $2B þð1 2 pÞ $5B ¼ $5B 2 $3B p: ð23:9Þ