offer that is less. So, if the buyer tenders an offer of five million dollars, then the dotcom owners will accept if their value is between zero and five million. The buyer, being
strategic, then realizes that this implies the value of the company to her
Fudenberg, Drew and Tirole, Jean (1991), Game Theory. MIT Press, Cambridge, MA.
Gibbons, Robert (1992), Game Theory for Applied Economists. Princeton University
Press, Princeton, NJ.
Myerson, Roger B. (1991), Game Theory: Analysis of Conflict. Harvard Uni
different results, and, ideally, each bidder would like to have access to all the surveys in
formulating its bid. Since the information is proprietary, that is not possible.
Strategic thinking, then, requires the bidders to take into account the additiona
information set, which has two nodes according to the different histories of play, which
player II cannot distinguish.
Because player II is not informed about its position in the game, backward induction
can no longer be applied. It would be better to sel
the potential buyers present, an auctioneer raises the price for the object as long as two
or more bidders are willing to pay that price. The auction stops when there is only one
bidder left, who gets the object at the price at which the last remaining op
While second-price sealed-bid auctions like the one described above are not very common, they provide insight into a Nash equilibrium of the English auction. There is a
strategy in the English auction which is analogous to the weakly dominant strategy in
A similar use of randomization is known in the theory of algorithms as Raos theorem,
and describes the power of randomized algorithms. An example is the well-known quicksort algorithm, which has one of the best observed running times of sorting algorithms
indifferent, receiving an overall expected payoff of 9 in each case. This can also be seen
from the extensive game in Figure 10: when in a weak position, player I is indifferent
between the moves Announce and Cede where the expected payoff is 0 in each ca
(20, 4)
stay
in
0.5
chance
(Announce) HH
H
H
sell HHH
I
H
out
II
(12, 4)
( 4, 20)
stay
in
@
@
HH
Announce
H
HH
0.5@
sell
HH
I
H (12, 4)
@
out
@
H
H
H
HH
Cede HH
H
(0, 16)
Figure 10. Extensive game with imperfect information between player I, a
Strategies in extensive games
In an extensive game with perfect information, backward induction usually prescribes
unique choices at the players decision nodes. The only exception is if a player is indifferent between two or more moves at a node. Then, an
developed the basis for a strong competing product. For brevity, when the large company
has the ability to produce a strong competing product, the company will be referred to as
having a strong position, as opposed to a weak one.
The large company, after
The payoffs in Figure 9 are derived from the following simple model due to Cournot.
The high, medium, low, and zero production numbers are 6, 4, 3, and 0 million memory
chips, respectively. The profit per chip is 12 Q dollars, where Q is the total quantit
high, medium, low, or none at all, denoted by H, M, L, N for firm I and h, m, l, n for
firm II. The market price of the memory chips decreases with increasing total quantity
produced by both companies. In particular, if both choose a high quantity of prod
correct comparison is to consider commitment to a randomized choice, like to a certain
inspection probability. In Figure 6, already the commitment to the pure strategy Inspect
gives a better payoff to player I than the original mixed equilibrium since pla
game. The analysis of dynamic strategic interaction was pioneered by Selten, for which
he earned a share of the 1994 Nobel prize.
First-mover advantage
A practical application of game-theoretic analysis may be to reveal the potential effects
of changing t
gies of the players. In the game tree, any strategy combination results into an outcome
of the game, which can be determined by tracing out the path of play arising from the
players adopting the strategy combination. The payoffs to the players are then en
II
H
(2, 2)
HH
H
dont HHH
H (0, 1)
buy
High
I
buy
*
@
@
@
buy
Low @
@
@
HH
II
H
(3, 0)
H
HH
j
dont HHH
H (1, 1)
buy
Figure 7. Quality choice game where player I commits to High or Low quality, and
player II can react accordingly. The arrows indicate the
lutionary game. Unlike Figure 5, it is a non-symmetric interaction between a vendor who
chooses Dont Inspect and Inspect for certain fractions of a large number of interactions.
Player IIs actions comply and cheat are each chosen by a certain fraction of
20, the payoff 90 in Figure 6 should be changed to 200. The units in which payoffs
are measured are arbitrary. Like degrees on a temperature scale, they can be multiplied
by a positive number and shifted by adding a constant, without altering the underlyi
them (that is, play them randomly) without losing payoff. The only case where, in turn,
the original mixed strategy of player I is a best response is if player I is indifferent. According to the payoffs in Figure 6, this requires player II to choose compl
Mixed equilibrium
What should the players do in the game of Figure 6? One possibility is that they prepare
for the worst, that is, choose a max-min strategy. As explained before, a max-min strategy
maximizes the players worst payoff against all possible c
@ II
I@
Dont
inspect
comply cheat
10
0
0
10
0
90
Inspect
1
6
Figure 6. Inspection game between a software vendor (player I) and consumer (player II).
is that the vendor chooses Dont inspect and the consumer chooses to comply. Without
inspection, the cons
Figure 5 shows the bandwidth choice game where each player has the two strategies
High and Low. The positive payoff of 5 for each player for the strategy combination
(High, High) makes this an even more preferable equilibrium than in the case discussed
ab
time, the same strategy names will be used for both players). For player II, High and Low
replace buy and dont buy in Figure 4. The rest of the game stays as it is.
The (unchanged) payoffs have the following interpretation for player I (which applies
in t
is easy to see that the critical fraction of High users so that this will take off as the better
strategy is 1/5. (When new technology makes high-bandwidth equipment cheaper, this
increases the payoff 0 to the High user who is meeting Low, which changes t
strategy combination that is not a Nash equilibrium is not a credible solution. Such a
strategy combination would not be a reliable recommendation on how to play the game,
since at least one player would rather ignore the advice and instead play another s
@ II
I@
dont
buy
buy
1
2
High
2
0
0
1
Low
1
1
Figure 4. High-low quality game with opt-out clause for the customer. The left arrow
shows that player I prefers High when player II chooses buy.
customer does not sign the contract in the first place, since
resulting outcome, seen from the flow of arrows in Figure 3, is still unique. Another
way of obtaining this outcome is the successive elimination of dominated strategies: First,
High is eliminated, and in the resulting smaller game where player I has only
Any two-player game in strategic form can be described by a table like the one in
Figure 1, with rows representing the strategies of player I and columns those of player II.
(A player may have more than two strategies.) Each strategy combination defines a
of service, High or Low. High-quality service is more costly to provide, and some of the
cost is independent of whether the contract is signed or not. The level of service cannot
be put verifiably into the contract. High-quality service is more valuable t