Incomplete Information: Part II
A Dynamic Game
Since we are dealing with a dynamic game, time plays an important role. Since time is important and we are considering games with incomplete information, we have to be concerned with who knows what and when they know it. Again we distinguish between imperfect and incomplete information. Imperfect information suggests that we may be making moves at the same time as our opponent or do not know our opponent's move until our own move has been made. Incomplete information refers to the possibility that our opponent's type may be unknown to us, and so we do not know her payoffs. 
Rules for Information Sets
Because it is so important to know who knows what and when they know it, we need to impose some rules on information sets.
Rule 1: All the nodes in an information set must belong to the same player. Any two players must have their own information sets.
Rule 2: If decision node A is a predecessor of decision node B, then A and B cannot be members of the same information set. You can't forget what you or other players have done in the past.
Rule 3: The same set of actions must be available at each node within an information set. Within an information set the member nodes must be logically indistinguishable. Suppose nodes A and B are members of the same information set. From A you can choose either 'up' or 'middle'. From B you can choose 'middle' or 'down'. If Gretl is at node A then she knows she can choose between 'up' and 'middle', and therefore she cannot be at node B. Consequently, A and B are not in the same information set.
Rules for Probabilities: A Review of Bayes Rule
Since chronology and the sequence of moves are important it may be that we can learn from events that precede the node at which we must make a decision. Recently Charlene has met Will. She knows that 10% of the male population has HIV. Her prior probability that Will is infected is 10%. There is a test for HIV and Will agrees to take it. It is known that among those with HIV, the test will come back positive 95% of the time. Among those who do not have HIV, the test returns a positive result 10% of the time. Will's test has come back negative. How can Charlene use this information?
Let's write down everything we know.
P stands for probability. The term in parentheses is an event. The statement is read as 'the probability that a randomly drawn male has HIV is 10%'.  
The probability that a randomly drawn male does not have HIV is 90%. The bar over the event HIV indicates that the event of interest is the complement to HIV, i.e., not having HIV.  
There are now two events in parentheses: Having HIV and testing positive for HIV. The vertical line is read as "given". The probability statement is read as 'Given that a randomly drawn male has HIV, the probability of testing positive is 95%'.  
Given that a randomly drawn male does NOT have HIV, the probability of testing positive is 10%. 
Notice in the two conditional probabilities we have preserved the sense of time: One believes that one may have contracted HIV, then one is tested.
Actually, we know two more pieces of information. means that in the pool of those with HIV, 95% will test positive. Therefore, in the pool of those with HIV, 5% will test negative. Or, .
Similarly, means that in the pool of those without HIV, 10% will test positive. Therefore, in the same pool of individuals without HIV, 90% will test negative. Or, .
From these numbers we can calculate some probabilities that may be of more interest to Charlene and Will. From the rules of probability we know that the intersection is equal to the product of the conditional and the corresponding marginal. In the notation that we have been using where the upside down U means either 'and' or 'intersection'. The probability statement is read 'The probability of testing positive and having HIV is 9.5%.' We can use this rule to compute the other intersections and summarize them, as usual, in a table:

HIV 

Test + 
.095 
.09 
Test  
.005 
.81 
We still aren't home. Charlene is interested in changing the order of temporal precedence. She wants to be able to revise her estimate of the probability that Will has HIV given the result of his test. That is, she is interested in . As a good student of probability, Charlene knows that she can use to find the answer to her question.
Given that Will's test has come back negative, there is a 99% chance that he is HIVfree. Suppose that his test has come back negative, what would be the probability that he would have HIV? There would only be a slim chance:
Suppose that his test has come back positive, what would be the probability that he would NOT have HIV? The result is perhaps surprising and counterintuitive.
If you are Charlene, what do you do about Will?
Perfect Bayesian Equilibrium
The Dating Game
Let's take a closer look at a stylized dating game between Will and Charlene. Will is one of two types. He either has HIV or he doesn't, and he knows which type he is. Charlene doesn't know his type, but she knows that there is a test that can reveal his type with a high degree of accuracy. Will asks Charlene for a date. Concerned about her own health, and not knowing Will's type, Charlene asks Will to have the HIV test before she will agree to go out with him. With that condition on getting a date with Charlene, Will must decide whether or not to have the test done. Even after Will decides what to do about the test, Charlene must decide whether or not she will date him or stay home. The game tree is pictured below, and you can get the Gambit game if you click here.
The player called chance determines whether Will has HIV. This is the Harsanyi transformation that was introduced in static games of incomplete information. The lengths of the black line segments in the branches coming off the chance node are proportional to the share of the population that has HIV. Under the rules of what constitutes an information set and what constitutes a subgame, we can see that there are no proper subgames.
This is a game of incomplete information since Will knows his type but Charlene doesn't. It is dynamic since Will agrees to an HIV test, then Charlene agrees to a date. Notice the structure of the information sets. Will's acceding to Charlene's request for the test doesn't tell her his type, but it does say something about his character.
If Charlene stays home then her payoff is always zero. If Will does have HIV, refuses the test, and Charlene agrees to the date then his payoff is 3 and hers is 1; he gets the date and she is at risk for the disease.
If Will has HIV, agrees to the test, and Charlene agrees to the date then their payoffs are (2, 1); Will gets the date and presumably Charlene is well enough informed to avoid risky behavior. If Will does not have HIV, declines the test, and Charlene stays home then their payoffs are (1, 0); Charlene's payoff is obvious. Will's payoff is 1 since he lost out on the date due to his own pigheadedness. The rest of the payoffs are reasoned in a similar manner. Before going further we put the payoffs in tabular form. This is NOT the strategic form of the game since it does not make use of the strategy profiles for Will and Charlene.
Will  
HIV  Not HIV  
Take Test  Refuse Test  Take Test  Refuse Test  
Charlene  Date  1 , 2  1 , 3  2 ,3  1 ,2 
Home  0 , 2  0 , 1  0 ,1  0 ,1 
The strategic form of the table is pictured below. Will's strategies tell him what to do for each of his two types. If he has HIV he can agree to the test and if he doesn't have HIV he can agree to the test. That is the first column under Will; (Test, Test). Each of Charlene's strategies has two moves. The first corresponds to her move when Will agrees to the test and the second corresponds to her move when Will refuses to take the test. For example, (Date, Date) tells Charlene to 'Date' if Will agrees to the test and 'Date' if he refuses the test. To help you keep things straight I have subscripted the first row and column strategies.
To determine the expected payoffs consider the first row and first column cell. Will always Refuses to Test and Charlene always dates. Her expected payoff is 1(1/10)+1(9/10). Will's payoff is 3 or 2 depending on whether he is HIV or not.
Will's second strategy is to Refuse if he has HIV and Test if he doesn't. Charlene's second strategy tells her to go out on a Date whenever she hears Refuse and stay Home whenever she hears Test. Charlene's expected payoff will be 1(1/10) + 0(9/10). Will's payoffs are either 3 or 1 depending on whether he has HIV.
Charlene  
Date_{Refuse}, Date_{Test}  Date_{Refuse}, Home_{Test}  Home_{Refuse}, Date_{Test}  Home_{Refuse}, Home_{Test}  
Will  Refuse_{HIV}, Refuse_{not HIV} 
(3,2), 4/5  (3,2), 4/5  (1,1), 0  (1,1), 0 
Refuse_{HIV}, Test_{not HIV} 
(3,3), 17/10  (3,1), 1/10  (1,3), 9/5  (1,1), 0  
Test_{HIV}, Refuse_{not HIV} 
(2,2), 1  (2,2), 9/10  (2,1), 1/10  (2,1), 0  
Test_{HIV}, Test_{not HIV} 
(2,3), 19/10  (2,1), 0  (2,3), 19/10  (2,1), 0 
Let's do one more cell. Will's third strategy is Test if he is HIV and Refuse if he is not. Charlene's third strategy is Stay Home if she hears Refuse and Date if she hears Test. For example, when Will has HIV he gets the Test and Charlene Dates. Her expected payoff 1(1/10)+0(9/10).
Date, Date dominates Date, Home and Home, Home for Charlene. Once you eliminate those two strategies you can see that Test, Test dominates Test, Refuse and Refuse, Test dominates Refuse, Refuse.
Continuing with IEDS you will find that the solution to the game is Test_{HIV}, Test_{not HIV} and Home_{Refuse}, Date_{Test}.
There is one more Nash equilibrium in the game at Refuse_{HIV}, Refuse_{not HIV} and Date_{Refuse}, Home_{Test}. Is one of the two equilibria better, and by what criterion?
For two other versions of the game right click on these links and open the file with Gambit: 1. In which Charlene sees the test result and must decide what to do. 2. In which Will agrees to and produces the test result and then Charlene decides what to do.
The Surfboard Game Again
The new wave in the game is that HangTen chooses to enter after CFoam has decided to expand. But upon seeing CFoam's new large plant, HangTen still doesn't know whether or not CFoam is high or low cost. The game tree is pictured below and you can click here to get the Gambit *.efg game file.
The strategic form of the game is
CFoam  
Expand_{High}, Expand_{Low}  Expand, Don't expand  Don't Expand, Expand  Don't Expand, Don't Expand  
HangTen  Enter_{Expanded}, Enter_{Didn't Expand}  1, (1,1)  1/3, (1,2)  1/3, (2,1)  1, (2,2) 
Enter, Stay out  1, (1,1)  1/3, (1,3)  2/3, (3,1)  0, (3,3)  
Stay out, Enter  0, (2,4)  2/3, (2,2)  1/3, (2,4)  1, (2,2)  
Stay
out, Stay out 
0, (2,4)  0, (2,3)  0, (3,4)  0, (3,3) 
Stay out, Enter is HangTen's dominant strategy. HangTen believes that Expand, Expand is dominated by Don't Expand, Expand for CFoam. Also Expand, Don't expand is dominated by Don't expand, Don't expand. From this HangTen is able to conclude that CFoam will never expand when their cost of doing so is high. Once CFoam's dominated strategy is eliminated, and keeping in mind HangTen's dominant strategy, it becomes apparent that CFoam will play Don't expand, Expand. The IEDS equilibrium, which is a Nash equilibrium, is highlighted.
There are other Nash equilibria in the game. Consider the following table of best responses:
HangTen  CFoam 
b(X, X) = (S,E) and (S,S)  b(S,E) = (X,X) and (D,X) 
b(X,D) = (S,E)  b(E,S) = (D,D) 
b(D,X) = (S,E)  b(S,S) = (D,X) 
b(D,D) = (E,E) and (S,E)  b(E,E) = (D,D) 
The other two Nash equilibria are highlighted in the normal form of the game. The other two Nash equilibria are not plausible. Does it make sense for CFoam to expand when the cost of doing do is high? Does it make sense for HangTen to enter no matter what, or for CFoam to never expand? Since there are no proper subgames, all three of the Nash equilibria are subgame perfect by default.
How would we resolve the problem of a multiplicity of Nash equilibria when we are unable to apply IEDS?
There are two nontrivial information sets, labeled (2,1) and (2,2) in the diagram. Information set (2,1) corresponds to HangTen observing that CFoam has built more capacity. Information set (2,2) corresponds to the observation that CFoam has not expanded. Before the game is played HangTen believes that there is a 2/3 chance that CFoam is low cost. HangTen might want to revise that prior once they have seen that CFoam has added capacity. Indeed, from the normal form of the game we concluded that CFoam would never expand if it was high cost. From the rules of probability HangTen would then find that P(LoExpand) = 1. HangTen's full set of revised probabilities would be called its belief profile.
This brings us to the definition of a perfect Bayes equilibrium:
1. The collection of strategies constitute a Nash equilibrium, given the player's beliefs.
2. At each player's information set, the move required by the player's strategy maximizes that player's expected payoff, given that player's beliefs and the other player's startegies.
3. Every player's beliefs can be derived from the equilibrium strategy profile and the common prior beliefs.
Neither {(Expand, Expand), (Stay out, Enter)} nor {(Don't expand, Don't expand), (Enter, Enter)} has a belief profile that would be part of a perfect Bayes equilibrium.
The methods of dynamic games with incomplete information have been used to study the market for lemons. Usually the terminology refers to the market for cars, principally used cars, but it can also be used to study labor markets and the market for bank loans.