# Principles of Game Theory Lecture 17: Pricing as a Mechanism Design problem.

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Principles of Game Theory Lecture 17: Pricing as a Mechanism Design problem

Administrative FCEs are up Current response rate = 0%. Final on Thursday Majority of material will be from content post-midterm Quiz results… It really wasnt meant to be difficult.

Quiz

Quiz - Separating 2 Separating Eq: 1.Sender: Type T1 plays R, Type T2 plays L; Receiver: If L, then play D. If R, then U. 2.Sender: Type T1 plays L, Type T2 plays R; Receiver: if R, then play U. If L, then D. Almost all of you did not specify the complete strategy of the Receiver. This is crucial because you must specify the complete contingent plan so can verify that the Sender doesnt have an incentive to deviate. Also most of you at most only listed one equilibrium but you should check all possible forms of separation.

Quiz - Pooling There is one pure strategy pooling equilibrium: Both types of Senders (T1 and T2) play L and the Receiver plays U upon observing either L or R. Why is R for both types of Senders and U for the Receiver not an equilibrium? Most people did get the Bayesian updating question.

Last time: Price Discrimination Price setting (in specific markets) can be thought of as a mechanism design problem in order to deal with Adverse Selection: Principle: Firm owner Agent: Consumer with different types of willingness to pay If the firm could know your willingness to pay, what would they charge? Once you start to look, youll see many screening devices set up to try to figure this out (eg: airline schedules)

Price Discrimination An example Suppose an airline is selling tickets to two types of consumers: business and tourist Doesnt know which type it is (eg; want to travel on the same day, each is booking via a website, etc) Each type of consumer has a different willingness to pay for tickets 70% Tourist 30% Business Airline can offer two types of ticket: first class and economy.

Airline Price Discrimination Assume 100 consumers (from the book: p546) What should the airline charge for each ticket? Type of Service Airlines Cost Reservation PriceAirline Profit TouristBusinessTouristBusiness Economy10014022540125 First Class15017530025150

Airline Price Discrimination: Ideal solution Ideal solution (baseline): if it could observe types Sell only First Class tickets to Business and only Economy to Tourists 150*30 +40*70 = \$7300 But first best isnt feasible – Airline doesnt observe type Type of Service Airlines Cost Reservation PriceAirline Profit TouristBusinessTouristBusiness Economy10014022540125 First Class15017530025150

Airline Price Discrimination: What would happen if firm naively priced at the first best solution? 140 for the Economy (for the Tourist) 300 for the First Class (for the Business Traveler) No Tourist would buy a First Class ticket but neither would a Business traveler (bad). Why? From the business consumers point of view, hes better off buying an economy ticket: 175-140 > 300-300 Type of Service Airlines Cost Reservation PriceAirline Profit TouristBusinessTouristBusiness Economy10014022540125 First Class15017530025150

Airline Price Discrimination So how do we solve the Airlines problem? Set up the right IC and IR constraints and optimize. Whats the tension (tradeoff)? Airline must keep the 1 st -class fare sufficiently low to give business travelers enough incentive to choose this service. Otherwise business travelers defect and choose economy tickets. 1 st class tickets must be incentive compatible for business Economy must be incentive compatible for tourists.

Constraints IC constraints: Tourists: surplus from economy must be greater than surplus from 1 st class: Business travelers: surplus from business must be greater than economy:

Constraints What about the Participation constraints? Traveling on the airline must be better for each type than not. Knowing the consumers outside option is important. (well normalize it to zero in this case).

Optimal Pricing Now the problem is just one of constrained maximization:

End of course. There are many more very cool topics left that one could study as a game theoretic analysis, but there simply isnt time. Delegation, collective action, bargaining, auctions, cooperative game theory, matching, market design, etc… (Honestly there wouldnt be time even if this course were 2 semesters long.) But Ive had a lot of fun teaching the course and hope youve enjoyed it – and gotten something out of it too. Remember: Ive tried to give you lots of little pieces of advice/takeaways. Think about incentives of everyone! Dont hate the player…

Exam Review Whats on it? Everything. I havent written it yet –my office door will probably be closed this afternoon except during office hours. But if I had to guess: 20-30% pre-midterm 10-20% repeated games 20-40% bayesian games 20-30% contract theory / mechanism design.

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