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### Rule 3

Lecture 3ADominance

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.
- They are also used in competitive contracting between an (auctioneer) firm and other (bidder) firms up or down the supply chain to reach trading agreements.
- Here we compare two sealed bid auctions, where there are two bidders with known valuations of 2 and 4 respectively.

First price sealed bid auction

- In a first price sealed auction, players simultaneously submit their bids, the highest bidder wins the auction, and pays what she bid for the item.
- Highway contracts typically follow this form.
- For example, if the valuation 4 player bids 5 and the valuation 2 player bids 1, then the former wins the auction, pays 5 and makes a net loss of 1.

Dominance in a first price auction

- There is a weakly dominant strategy for the row player to bid 1.
- Eliminating all the other rows then leads the column player to maximize his net earnings by bidding 2.

Second price sealed bid auction

- In a second priced sealed bid auction, players simultaneously submit their bids, the highest bidder wins the auction, and pays the second highest bid.
- This is similar to Ebay, although Ebay is not sealed bid.
- For example, if the valuation 4 player bids 5 and the valuation 2 player bids 1, then the former wins the auction, pays 1 and makes a net profit of 3.

Dominance in a second price auction

- The row player has a dominant strategy of bidding 2.
- The column player has a dominant strategy of bidding 4.
- Thus both players have a dominant strategy of bidding their (known) valuation.

Comparing two auction mechanisms

- Comparing the two auctions, the bidder place different bids but the outcome is the same, the column player paying 2 for the item.
- How robust is this result, that the form of the auction does not really matter?
- The first part of the strategy course sequel 45-975, Auction and Market Strategy, analyzes this question in depth, and more generally, investigates optimal bidding and auction design.

Why might outcomes depend on the way games are presented?

- The difference between the outcomes in the strategic form versus the extensive forms is remarkable.
- There are reasons why subjects “stay open” in the extensive form and “close shop” in the strategic form:
- Subjects are risk lovers, and like gamblers are willing to pay for the opportunity to gamble with nature.
- Subjects are confused by the calculations required to maximize expected value.

The field manager

- If a field manager can fool his supervisor, then fabrication maximizes his compensation and career prospects. But his worst outcome is to get caught.
- Also the field manager is given more credit from the firm if his supervisor conducts a diligent review of a sound business proposal, than if his supervisor does not thoroughly review it.

The regional supervisor

- The regional supervisor is rewarded by the firm when he detects problems in the field, and also when his field manager makes sound business proposals.
- If the supervisor is not diligent he cannot detect self serving behavior, but he can recognize sloth.
- Also diligent checking interferes with his other activities at work and home.

Supervision

- Games with dominated strategies do not necessarily have dominant strategies.
- Here we see to “Propose a least effort alternative” is dominated by “Diligently create . . .”

Each player should discard his dominated strategies

Marketing groceries

- In this 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 > -1/3

π50+(1-π)55 > 52 or 3/5 > π

π60+(1-π)50 > 55 or π > ½

- Hence all mixtures of π satisfying the inequalities:

½ < π < 3/5

dominate the hours strategy.

Eliminating a dominated strategy

Upon eliminating the hours strategy from the game, we see that a dominant strategy for the corner store emerges, that is choosing “hours”.

Killington

- In this game, an MBA student can either study on the weekend or, if the resort is open, ski.
- The resort, Killington, may decide to stay open even if rain turns the slopes to mud and ice.

Rationalizing the payoffs for the ski resort game

- The MBA student prefers to skiing to study if it snows, but prefers study to skiing if it rains.
- Given her preference ordering, one can prove that the solution of the game is not affected by the values the MBA student places on each outcome.
- Killington’s profits whenever the MBA student skis, but makes higher profits if it snows.
- Killington makes losses if they open and the student studies. Those losses are smaller if it snows, because its employees have the slopes to themselves.

Strategies in the ski resort game

- Killington’s strategies are to:
- open
- close
- open only if it snows
- open only if it rains
- The MBA’s strategies are to:
- ski
- study

Payoff calculations

For each strategy pair and corresponding matrix cell, we compute the expected payoffs using the probabilities of rain versus snow.

Strategic form of ski resort game

- The strategic form for this game is easier to analyze than its extensive form.
- The bottom strategy of Killington is dominated by the one above it.

Iterating further

- If the MBA believes Killington does not play dominated strategies, then he would eliminate the bottom strategy from consideration, revealing a dominant strategy to “ski”.
- If Killington knows the MBA will reason in this fashion, then its best response is to stay open regardless of the whether.

Lecture summary

- Some games are easier to analyze when presented in their strategic form than in extensive form.
- We derived a third rule that applies to the strategic form: do not play dominated strategies.
- Our experiments also suggest that we might extend the dominance principle. If a player recognizes that another player will apply Rules 2 and 3, this may simplify the game for her.

Lecture 4AIterative Dominance

This lecture continues our study of the strategic form, extending the principle of dominance to iterative dominance.

Market games

- Our next pair of examples illustrate how the strategy space can greatly affect the profitability of firms competing in a concentrated industry.
- Suppose there are just two firms in the industry. We shall see that their market value depends on whether they compete on price, or on quantity.

Demand and Technology

- Consumer demand for a product is a linear function of price, and that market pre-testing has established:
- We also suppose that the industry has constant scale returns, and we set the average cost of producing a unit at 1.

Price competition

- When firms compete on price, the firm which charges the lowest price captures all the market.
- When both firms charge the same price, they share the market equally.
- These sharp predictions would be weakened if there were capacity constraints, or if there was some product differentiation (such as location rents or market niches).

Profit to the first firm

- As a function of (p1,p2), the net profit to the first firm is:
- Net profit to the second firm is calculated in a similar way.

Market games with price competition

- In our example q = 13 – p and c= 1.
- We could try to solve the problem algebraically.
- An alternative is to see how human subjects attack this problem within an experiment.
- We have substituted some price pairs and their corresponding profits into the depicted matrix.

Solving the price setting game

- Setting price equals 7 is dominated by a mixture of setting price to 5 or 2, with most of the probability on 5.
- Eliminating price equals 7 for both firms we are left with a 3 by 3 matrix.
- Now setting price equals 5 is dominated by a mixture of setting price to 3 or 2.
- In the resulting 2 by 2 matrix a dominant strategy of charging 2 emerges for both players.

Quantity competition

- When firms compete on quantity, demanders set a market price that clears inventories and fills every customer order.
- If firms have the same constant costs of production, and hence the same markup, then their profits are proportional to their market share.
- This predictions might be violated if the price setting mechanism was not efficient, or if the assumptions about costs were invalid.

Calculating profits when there is quantity competition

- Letting q1 and q2 denote the quantities chosen by the firms, the industry price is derived from the demand curve as:

p = ( - q1 – q2)/ = 13 - q1 – q2

- When the second firm produces q2, as a function of its choice q1, the profits to the first firm are

q1( - q1 – q2)/ - q1c = q1[12 - q1 – q2]

- The profits of the second firm are calculated the same way.

Market games with quantity competition

- As in the price setting game, we could try to solve the game algebraically, or set the model up as an experiment.
- If we can compute profits as a function of the quantity choices, using the second approach, we can easily vary the underlying assumptions to investigate the outcomes of alternative formulations.

Solving the the quantity setting game

- For both firms, setting quantity equals 6 is dominated by setting quantity equals 5
- Eliminating the strategy of choosing 6 for both firms, we are left with a 3 by 3 matrix in which the weakly dominant strategy is to pick quantity equals 4.

Iterative dominance

- Rules 2 and 3 rely on a player recognizing strategies to play or avoid independently of how others behave.
- If all players recognized situations in which these two rules applied and abided by them, and one of the players realized that, then this particular player should exploit this knowledge to his own advantage by refining the set of strategies the other players will use.
- Knowing which strategies the other players have eliminated reduced the dimension of his problem, ruling out possible courses of action that might otherwise look reasonable.

Is the algorithm of iteratively removing dominated strategies unique?

- Question: Can we have different solutions if we use different sequence of truncations?
- Answer: No
- Fact: Different algorithms for eliminating strictly dominated strategies lead to the same set of solutions.
- The key to proving this point is that if a strategy is revealed to be dominated it will remain dominated if another strategy is removed first.

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.

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).

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) = 4.0

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.

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 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?

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.

Four rules for good strategic play

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.

Lecture summary

- The second two rules, “play dominant strategies” and “do not play dominated strategies”, apply independently of whether the other players are rational or not.
- In this lecture we advocated using a fourth rule that applies to the strategic form: “iteratively eliminate dominated strategies”.
- Like our first rule, “look forward and reason back”, the fourth rule assumes that the other players are rational. In this case we are assuming that they will also apply the fourth rule for their own purposes.

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