Incomplete Information

This lecture shows how to analyze games that have more complicated information sets than complete information games. This requires us to apply rules, already learned in previous lectures, to more complicated settings.

Incomplete information games

All games are either complete information games or incomplete information games.

Therefore simultaneous move and perfect information games are not games of incomplete information.

A game has incomplete information if one player, when making her move, does not know information that would affect her choice that was known by another player when he had moved at an earlier point in the game, that is at a predecessor node.

Read Chapter 12, “Incomplete Information Games” in Strategic Play.

Defining incomplete information games

Suppose one player called Alpha moves before another player Beta does. That is Alpha’s decision node precedes Beta’s.

If Alpha is more informed than Beta about an event that occurred before either of them had the opportunity to move, then they are playing a game of incomplete information.

Alpha might wish to signal to Beta his knowledge through his move, or he might wish to confuse Beta, depending on their strategic complementarities.

What kind of information?

Incomplete information can arise from two sources:

1. Knowledge about a player’s past move:For example Gamma moves first, followed by Alpha and then Beta. Alpha sees Gamma’s move, Beta sees Alpha’s move, but Beta does not see Gamma’s move.

2. Knowledge about an earlier move by nature:Change the name of Gamma to nature in the above example, and let nature’s move be a random variable.

Incomplete information induced by earlier moves of other players

First we shall analyze two games in which one player knows more about a second player’s previous move than a third player.

Predators

This game features a corrupt government, a poorly run state enterprise and an opportunistic foreign investor wrestle for mineral and oil wealth.

Why Predators is anincomplete information game

When the foreign investor decides near the end of the game whether to withdraw or commit, he does not know whether the state enterprise has is trying steal his assets or not.

However if the government moves at all, it moves before the investor’s makes his final decision, and in that case knows that the state enterprise has indeed decided to try stealing the foreigners assets.

Governments seize assets given the opportunity

A dominant strategy for the government in this game is to seize foreign assets when presented with the opportunity to do so.

The reduced game upon anticipating the government’s choice

We now reduce the game by conditioning on the government’s dominant strategy.

If the government seizes its assets, the foreign firm should withdraw if it can.

If its assets were left intact, it should commit.

A further reduction of the predator game

Folding back the final decisions of the foreign investor, the further reduction yields a simultaneous move game.

Note the state owned enterprise has a dominant strategy to propose a joint venture.

Quality controlManufacturers do not consistently produce flawless products despite legions of consultants who have advised them against this policy.

Retailers help guard against flawed products by returning some of the defective items sent, and lending their brand to the ones they retail.

Consumers cannot judge product quality as well as retailers and producers, since each one experiences only a tiny fraction of the end product.

What is an acceptable defect rate, how often should retailers return defective items, and what are the implications for consumer demand?

Total quality management

If the consumer always buys . .

To solve this game we first remark that the consumer plays a mixed strategy in equilibrium. This claim can be established by contradicting the alternative hypothesis that she plays a pure strategy.

Suppose she always buys the product.

Then the retailer would never return a defective one, and the producer would specialize in producing defective products.

Thus the consumer only purchases defective products, and this is not a best response to the strategies of the manufacturer and the retailer.

If the consumer never buys . . .

But if the customer never buys the product, the retailer would always return defective ones.

In this case the manufacturer specializes in produced flawless products.

It now follows that the strategy of not buying is not a best response

Therefore the consumer follows a mixed strategy.

Defining the probabilities in the TQM problem

Let q denote the probability that the retailer offers a defective product item sale. Let r denote the probability the shopper buys the item. Let p be the probability a producer produces a flawless item.Both probabilities are strictly positive. receiving both kinds of products is strictly positive. If the shopper mixes between buying and not buying the product, then she must be indifferent between making either choice.

A schematic

Solving for r, the probability of buying

If 0 < q < 1, then the retailer is indifferent between offering a defective product and returning it.

In that case:

3r - 2(1 - r) = -1

⇒ 3r – 2 + 2r = -1

⇒ 5r = 1

⇒ r = 0.2

How to solve for p and q

Once we substitute for r = 0.2 in the shopper’s decision, we are left with the diagram:

1. q is chosen so that the producer is indifferent between production methods;

2. p is chosen so that the shopper is indifferent between buying and not buying.

Solving q,the probability of offering the product

The producer will only mix between defective and flawless items if the benefit from both are equated:

[6r + (1 - r)]q - 3(1 - q) = [3r + (1- r)]

⇒ 2q – 3 + 3q = 1.4

⇒ 5q = 4.4

⇒ q = 0.88

Solving for p, the probability of producing a flawless product

Investigating the cases above shows that in a mixed strategy equilibrium r = 0.2 and q = 0.88.

Since the shopper is indifferent between buying the item versus leaving it on the shelf, there are no expected benefits of acquiring the item:

9p - 10(1 - p)q = 0 ⇒ (9 +10q)p = 10q

⇒ p = 44/89

Offering a partial refund

We now modify the game slightly. If the customer buys a defective product, she receives partial compensation.

A different outcome

In this case the manufacturer has a weakly dominant strategy of specializing in the production of flawless goods.

Recognizing this, the shopper picks a pure strategy of buying.

Realizing that the shopper will buy everything she is offered, the retailer never returns its merchandise to the manufacturer (and indeed there is never any reason too).

Lecture summary

We defined games of incomplete information as the complement of complete information games.

Then we explained how games of incomplete information might arise.

The tools used to solve simultaneous move and perfect information games can be combined to solve games of incomplete information.

We analyzed two examples of incomplete information generated within a three player game.

Incomplete Information due to Nature

This lecture considers four games where nature is the source of the incomplete information, and discusses the costs and benefits of making players more informed.

Starting a new business

In this game the entrepreneur reaches one of three information nodes before he decides whether to approach a potential business partner or not, but the partner does not know which node the entrepreneur reached.

Seeking outside funding for a new business is tricky because the entrepreneur seeking support typically knows much more about the probability of success than the angel or the venture capitalist.

Folding back from nature’s nodes

The analysis of this game can be somewhat simplified by investigating the reduced game:

The strategic form

There are 8 strategies for the entrepreneur, listed as rows in the matrix and 2 strategies for the potential partner (the columns).

Calculating the expected payoffs for each strategy pair we obtain the strategic form for this game.

Dominated strategies for the entrepreneur

Inspecting the entrepreneur’s strategies we see that the fourth and the eighth are dominated by the second and the sixth respectively.

Similarly mixing the second with the sixth strategies rule out the third and fifth.

Dominance solvable

Once strategy eight is ruled out, “Decline” is a dominant strategy for the potential partner.

The entrepreneur goes solo, unless there is nothing to be had.

Cost of incomplete information

The expected value of this game, the sum of the expected values to each player, from playing the solution is:

0.1 * 8 + 0.5 * 1.6 = 1.6

In a perfect information game, where the partner is as equally informed as the entrepreneur, it is easy to prove the expected value of the game is:

0.1 * (20 +10) + 0.5 * 1.6 = 3.8

Moreover the expected value of maximizing the sum to both players is:

0.1 * (20 +10) + 0.5 * (8 - 2) = 6

The alternative of corporate research

This solution to this game illustrates the problem that practically every independent inventor faces in funding his research for commercial development.

One alternative is for large companies to conduct research in-house, where scientists and inventors are paid a fixed wage to develop new products with company equipment and support, the profits from their discoveries being distributed as dividends to shareholders.

This approach creates two other problems:1. Workers resign from the company start their

own firms just before they announce major discoveries

2. Researchers on wages lack motivation.

Aircraft MechanicHiring expert help, when you are the residual claimant, is also fraught with costs associated with incomplete information.

We examine this question from the viewpoint of a company’s ultimate status symbol, its executive jet fleet.

Background log on Aircraft Mechanic

The flight crew notice the turbines emit a new noise when they start to turn over, although it disappears before executive management boards, so the passengers are unaware of any differences.

Executive personnel decide whether to have the plane checked into unscheduled maintenance, at a cost of 200.

If the problem is severe there will be an additional cost of 800 in parts plus 400 labor. If left unchecked, and the problem is severe, an extra 800 will later be spent in parts.

If the plane is checked, the (outsourced) mechanics advise personnel whether the noise signals whether the engines require new parts or not.

Aircraft mechanic reduced

Folding back the resolution of uncertainty that occurs if personnel ignores the noise, we see the expected loss to the firm from taking no action is 1000, and the expected payout to the mechanic is 200.

Should personnel have the plane checked?

Aircraft mechanic: Strategic form for subgame

If personnel has the plane checked, the following subgame ensues, which we present in strategic form.

Note the strategy “Authorize” is dominated by “Ignore”.

Best replies in thereduced strategic form

Only two cells do not have at least one arrow leading out.

They are the only pure strategy Nash equilibrium.

Both involve the Personnel choosing ignore.

In both equilibrium the expected payoffs are the same.

Cost of incomplete information

The expected cost to Personnel from having the plane checked is 1200, because the mechanic’s advice is sought and then ignored.

Therefore the solution to the game is not to have the plane checked.

An alternative arrangement to owning a corporate fleet is to outsource executive air travel, by leasing, “partial sharing”, or relying on an air-taxi service that owns and maintains its own fleet.

In the latter case the executive air-taxi service would solve a perfect information game with its own mechanics using backwards induction.

Air-taxi service

The solution to this game yields the Air taxi service losses of 800 and benefits for the mechanic of 400.

Personnel should be willing to pay Air taxi up to 200 rent to rid itself of its incomplete information problem, while the mechanic should be willing to take a wage cut of 200 to get the extra business.

Product development raceOften companies do not know precisely how much competition they will face before launching a new product:

Strategic form ofproduct development race

Both firms have a dominant strategy to advertise the product, which determines the unique solution to this game.

An industry newsletter

Now suppose a newsletter is produced to keep firms abreast of the latest developments. The extensive form becomes:

Sub-games

There are three proper sub-games beginning at nodes 2, 3, and 4.

If Thompson is the only firm to develop the product, it should advertise rather than choose a low price, and similarly for Smith.

The sub-game starting node 4, when both firms develop the product, illustrates the prisoners’ dilemma. The unique solution is for both firms to charge the low price.

The strategic cost of better information

This example shows that more information about an industry could sometimes hurt it.

Additional information helps firms to identify situations where their positions are opposed to each other, and induce competition that might lead to the detriment of all firms.

Industrial espionage

In this version of the game thompson has implanted a spy in smith who cannot steal the technology, but keeps thompson abreast of news about smith’s product development.

How should Smith behave? How does this compare with the solution(s) where there is no spy?

What happens if only one firm can acquire the information about its rival, with the new payoffs shown?

Medical malpractice

One problem health insurance providers face is fraudulent behavior by doctors who prescribe treatment for healthy clients.

Consider the following extensive form game:

Strategic form of medical malpractice

Solving, the patient takes the treatment with probability 5/6, while the doctor prescribes treatment 1/4 of the time to healthy patients.

Thus healthy patients receive the treatment with probability 5/24, about 20% of the time.

In the strategic form of the game, we see that “ignore” is a dominated strategy.

Furthermore the best replies indicate that the game has a unique mixed strategy Nash equilibrium.

Diagnostic test

Is it profitable to administer a test that verifies whether someone is ill or not?

Solving the perfect information game, the doctor will only prescribe treatment to sick patients, who will always take it.

Gains from testingThe solution to the perfect information game yields an expected benefit of 84 to the patient and 6 to the doctor.

In the malpractice game, the expected benefit to the patient is:

(3*82 + 50 + 15*84 + 5*44)/6*4 = 74

while the expected benefit to the doctor is:

(3*2 - 38 + 15*6 + 5*14)/ 6*4 = 5.3

Together the patient and doctor are willing to pay up to 10.6 to administer the diagnostic test.

Lecture summary

We analyzed several games of incomplete information that are induced by chance events, that is beyond the control of the players.

We showed that firm boundaries and its core competencies are partly defined by it’s the strategic implications of its knowledge base.

We also proved that in a strategic context the old adage that more information is better is fallacious.

Finally we proved that if ethical principles are widely held and acknowledged profitability and consumer satisfaction is increased. In our example a diagnostic test preventing medical malpractice served the same purpose.