Lecture 3 Iterated Dominance The Strategic Value of Information

Lecture 3 Iterated Dominance

The Strategic Value of Information

We define a strategy for a player, the sets of strategies available to a player for a game, and the strategic form. Then we explain the meaning of a dominant strategy and iterated dominance. The latter parts of this lecture analyze the value of information in strategic contexts. The intuition from decision theory, that more information cannot hurt, does not easily extend to strategic situations, because the anticipation and reactions of the other players must be accounted for.

Strategies

How general is the idea that the best response may not depend on what other players are doing?

To answer this question we introduce and define the concept of a strategy.

A strategy is a full set of instructions to a player, telling her how to move at all the decision nodes assigned to her.

Strategies respect information sets: the set of possible instructions at decision nodes belonging to the same information set must be identical.

Strategies are exhaustive: they include directions about moves the player should make should she reach any of her assigned nodes.

Another way of representing games

Rather than describe a game by its extensive form, one can describe its strategic form.

The set of a player’s (pure) strategies is called the strategy space. The strategic form of the game is a list of all the possible pure strategies for each player and the (expected) payoffs resulting from them.

Suppose every player chooses a pure strategy, and that nature does not play any role in the game. In that case, the strategy profile would yield a unique terminal node and thus map into payoffs.

Note that no information is lost when transforming a simultaneous move game from its extensive to its strategic form.

Strategic formTo summarize: the strategy space is the set of mutually exclusive collectively exhaustive (MECE) strategies.

The strategic form representation is less comprehensive than the extensive form, discarding detail about the order in which moves are taken.

The strategic form defines a game by the set of strategies available to all the players and the payoffs induced by them.

In two player games, a matrix shows the payoffs as a mapping of the strategies of each player. Each row (column) of the table corresponds to a pure strategy. The cells of the table respectively depict the payoffs for the row and column player.

Dominant strategies

Strategies that are optimal for a player regardless of what the other players do are called dominant.

Although a player's payoff might depend on the choices of the other players, when a dominant strategy exists, the player has no reason to introspect about the objectives of the other players in order to make his own decision.

Similarly when a dominant strategy exists, the player does not need to know the behavior of the other players to form his or her best response to the probability distribution characterizing their choices.

Investment broker

This game is neither a simultaneous move game, nor a perfect information game.

However the second mover, the client, knows more than the first mover, her broker.

Strategic form of this gameThe easiest way of solving this game is to directly analyze its strategic form.

The strategies for each player are shown in the matrix.

To obtain the payoffs suppose, for example, the broker chooses “tech” and the client’s strategy is “continue with broker”. Then the broker’s expected compensation is: 0.5*3 + 0.5*9 = 6

Solution to investment broker

The broker should choose “tech” because it is a dominant strategy.

If the investor recognized that the broker had a dominant strategy, the investor would use the signal she receives about the economy, picking the strategy “continue if new, liquidate if bubble”.

In that case she would be using the principle of iterated dominance to solve the game.

Predators

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

Strategy Space of the Players

The enterprise has two strategies:1. Propose joint venture2. Steal foreign expertise

The government has two strategies:1. Endorse partnership 2. Seize assets

The corporation has five strategies:1. Infiltrate and commit2. Infiltrate and withdraw3. Infiltrate and commit unless government

seizes4. Infiltrate and withdraw unless government

seizes5. Negotiate in good faith

Government will seize assets if 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.

Just in time

3/20*50+17/20*50=50

3/20*400-17/20*60=9

3/20*2+17/20*2=2

3/20*248-17/20*12=27

Reduced game for component supplier

Taking the expected value over the nodes that involve nature (and the possibility of breakdown), we obtain the reduced game depicted on the right.

The subgame reached by Pratt & Whitney choosing “Wait” and Boeing choosing “Wants part” is itself a perfect information game, and trivially solved by the choice “Make part”.

Strategic form for the reduced game

Folding the solution of the subgame into the extensive form, we see the resulting is a 2 by 2 simultaneous move game with the strategic form depicted.

MBA market

CMU, Pitt and Duquesne compete in their MBA evening programs, drawing from an overlapping demand pool.

Their reputations and cross synergies with other programs effectively shape the kinds of choices they offer.

One of the players has a dominant strategy, and the game can be solved using iterative dominance.

Dominated strategies

If a player has a dominant strategy, then her other strategies are called dominated.

More generally, a strategy is called dominated if there exists some other strategy yielding a higher expected payoffs regardless of the strategies that the other players pick.

Thus a person should never play a dominated strategy. She can earn more by choosing another strategy without knowing the choices of the other players in the game.

Marketing groceries

In this simultaneous move game the corner store franchise would suffer greatly if it competed on the same feature as the supermarket.

This is illustrated by the fact that its smallest payoffs lie down the diagonal.

Strategies dominated by a mixture

The supermarket's hours strategy is dominated by a mixture of the price and service strategies.Let π denote the probability that the supermarket chooses a price strategy, and (1-π) denote the probability that the supermarket chooses a service strategy. This mixture dominates the hours strategy if the following three conditions are satisfied:

π65+(1-π)50 > 45 or π > -1/3π50+(1-π)55 > 52 or 3/5 > ππ60+(1-π)50 > 55 or π > ½

Hence all mixtures of π satisfying the inequalities: ½ < π < 3/5

dominate the hours strategy.

Revisiting “look ahead and reason back”

Let us reconsider the first rule we derived of “look ahead and reason back” for games of perfect information.

At the bottom node aren’t we just eliminating those strategies that are dominated by any strategy that picks out the best move at the very end of the game?

This insight leads us to the following theorem, which can be easily proved by an induction: Games of perfect information can be solved by writing down their strategic form and applying Rule 3 .

So let us rewrite Rule 3 in terms of iterated dominance:

Rule 3 (revised)

All players should simplify the game by iteratively discarding dominated strategies.

Three rules for strategic play

Rule 1: Play your best response to the population distribution of conditional choice probabilities.

Rule 2: If there is a dominant strategy, play it. Rule 3: Iteratively eliminate dominated

strategies. (Note that Rule 3 covers look ahead and reason back.)

How sophisticated are the players?

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.

Why is dominance more compelling than iterated dominance?

If you don’t know the choice probabilities of the other players, the two most important principles for strategic play, are to play dominant strategies, and iteratively eliminate dominated strategies.

The first principle applies regardless of whether the other players are rational or not, and therefore does not depend on whether you know their payoffs or not.

The second principle applies when you know enough about the payoffs of the other players to recognize their dominated strategies.

The strategic value of informationIn decision theory, new information is never harmful and has positive value if it leads you to change your behavior.

The rules we have developed for solving games can be used to value information in strategic contexts. We ask:

1. Is there value form withholding information for the competition, or should we release it?

2. Does providing (the same) new information to everyone increase value?

3. What can we learn from the choices of others when they are more informed than us?

A follower’s advantage

Through orders bookings and sales, first entrants typically learn about potential demand earlier than later entrants.

If these data cannot be kept confidential, then followers can use the data.

Over on the right we see that Eagle decides whether to enter or not, only after seeing what Cheetah has done and the effects on demand.

Folding back and simplifying the game tree

If Cheetah begins an air service then Eagle will enter only if demand is high.

If Cheetah does not create the service, then Eagle will not get the information on demand.

In that case we can exchange the order of the moves of Eagle and nature.

A further reduction

Taking expected values we are left with a very simple game tree.

Cheetah should stay out, and Eagle should enter.

The value of withholding data on demand

Now suppose Cheetah can prevent Eagle from having access to data on the profitability of its new route.

In this case Eagle can see whether Cheetah entered or not, but not the state of demand.

Air service -redrawn

The game is equivalent to the picture on the right.

Both firms must move before the state of demand is revealed.

Air service – further reduction

Taking the expectation over the payoffs yields a further simplification.

Now Cheetah will enter confident that Eagle will stay out.

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:

Subgames

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 and the value of information silos

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.

Finally suppose Smith and Thompson were two plants owned by the one firm! This example shows the value of having information silos when different profit centers are competing with each other.

Bottling wine

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.

Extensive form game

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.

Strategies for Mondavi

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).

cork screwtop indifferent

y y y

y y n

y n y

n y y

y n n

n y n

n n y

n n n

Eliminating the dominated strategies of Mondavi

We can simplify the problem that Jarvis has by drawing its decision problem when Mondavi follows its dominant strategy.

Solving for Jarvis

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) = 4.0while 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.

Rivals as a source of information

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 noisy signal 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?

The value of independent research

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.

Summary

We developed the strategic form of a game, and demonstrated that some games are easier to analyze in their strategic form than in their extensive form. We analyzed a third principle for strategic play, iterative dominance, and showed that it encompasses backwards induction as an application. Then in the latter parts of this lecture we showed that the effects of new information are much more complicated to analyze in strategic situations than in decision theory.For example, we showed that when new information is provided to every player, they can all lose, while another example highlighted the benefits of withholding information.