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 real
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.
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,
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 induct
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
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
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 o
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 Anno
(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
[email protected]
sell
HH
I
H (12, 4)
@
out
@
H
H
H
HH
Cede HH
H
(0, 16)
Figure 10. Extensi
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 in
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 positi
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
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
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
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 analys
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 strat
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
pl
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 com
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 shif
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 payof
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
ma
@ II
[email protected]
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 c
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 mo
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 fo
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
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 wo
@ II
[email protected]
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.
custome
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, an
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 th
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 int