Lecture 3A Dominance. This lecture shows how the strategic form can be used to solve games using the dominance principle. Auctions. Auctions are widely used by companies, private individuals and government agencies to buy and sell commodities.
This lecture shows how the strategic form can be used to solve games using the dominance principle.
Each player should discard his dominated strategies
π65+(1-π)50 > 45 or π > -1/3
π50+(1-π)55 > 52 or 3/5 > π
π60+(1-π)50 > 55 or π > ½
½ < π < 3/5
dominate the hours strategy.
Upon eliminating the hours strategy from the game, we see that a dominant strategy for the corner store emerges, that is choosing “hours”.
For each strategy pair and corresponding matrix cell, we compute the expected payoffs using the probabilities of rain versus snow.
This lecture continues our study of the strategic form, extending the principle of dominance to iterative dominance.
p = ( - q1 – q2)/ = 13 - q1 – q2
q1( - q1 – q2)/ - q1c = q1[12 - q1 – q2]
Applying the principle of iterative dominance assumes players are more sophisticated than applying the principle of dominance.
Applying the dominance principle in simultaneous move games makes sense as a unilateral strategy.
In contrast, a player who follows the principle of iterative dominance does so because he believes the other players choose according to that principle too.
Each player must recognize all the dominated strategies of every player, reduce the strategy space of every player as called for, and then repeat the process.
Corks are traditionally used in bottling wine, but recent research shows that screwtops give a better seal, and hence the reduce the risk of oxidation and tainting. They are also less expensive.
However consumers associate screwtops with cheaper varieties of wine, so wineries risk losing brand reputation from moving too quickly ahead of the consumer tastes.
To illustrate this problem consider two Napa valley wineries who face the choice of immediately introducing screwtops or delaying their introduction.
Mondavi has resources to conduct market research into this issue, but Jarvis does not.
However Jarvis can retool more quickly than its larger rival, so it can copy what Mondavi does.
A strategy for Mondavi is whether to introduce screwtops, abbreviated a “y”, or retain corking, abbreviated by “n”, for each possible triplet of consumer preferences.
Therefore Mondavi has 8 different strategies.
Reviewing the payoffs in the extensive form, the unique dominant strategy for Mondavi is (n,y,y).
We can simplify the problem that Jarvis has by drawing its decision problem when Mondavi follows its dominant strategy.
Since 4 > 0, Jarvis bottles with cork if Mondavi does.
The expected value of using screwtops when Mondavi does is:
(0.3*4 + 0.2*4 )/(0.2 +0.3) = 4.0
while the expected value of retaining corking when Mondavi switches is:
(0.3 + 0.2*6)/(0.2 +0.3) = 3.0
Therefore Jarvis always follows the lead of Mondavi.
The solution to this game shows that rivals can be a valuable source of information.
Although Jarvis could undertake its own research into bottling, it eliminates these costs by piggybacking off Mondavi’s extensive marketing research.
Nevertheless Jarvis receives a noisysignal from Mondavi. Jarvis cannot tell whether consumers prefer screwtops or are indifferent.
How much would Jarvis be prepared to pay to conduct its own research, and receive a clear signal?
When consumers are indifferent Jarvis could capture a niche market by corking, increasing its profits by 6 – 4 = 2.
Hence access to Mondavi’s superior market research increases Jarvis’s expected net profits by:
0.2*2 = 0.4.
This sets the upper bound Jarvis is willing to pay for independent research.
Rule 4: Iteratively eliminate
Rule 1: Look ahead and reason back
Rule 2: If there is a dominant strategy, play it
Rule 3: Discard dominated strategies.
Rule 4: Iteratively eliminate dominated strategies.