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Seminar in Auctions and Mechanism Design Based on J. Hartline’s book: Approximation in Economic Design Presented by: Miki Dimenshtein & Noga Levy

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1. Introduction to multi-dimensional environments. *Item pricing against m-agents auction. 2. Reduction to single dimensional preferences. 3. Lottery pricing. Lecture topics

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Background In previous chapters, we’ve consider agents' types as single dimensional - agent’ private preference is given by a single value for receiving a service We now turn to multi-dimensional environments where the agents’ preferences are given by a multi- dimensional type Example – multi-dimensional environment a home buyer may have distinct values for different houses on the market.

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Consider selling m items and n agents Each agent has a valuation function that is defined across all bundles. I.e., if agent i receives bundle S ⊂ {1,...,m} then he has value v i (S). A generally multi-dimensional auction assigns to agent i, bundle S i and payment p i. Agent i’s utility is given by vi(S i ) − p i. This called quasi-linear. The mechanism chooses the outcome that maximizes social surplus and it charges each agent the externality imposed on the remaining agents generally multi-dimensional

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Theorem For agents with (generally multi- dimensional) quasi-linear preferences, the surplus maximization mechanism is dominant strategy incentive compatible and maximizes the social surplus.

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We will show that posted-pricing mechanisms can approximate the optimal social surplus in some relevant environments (not all of them). Even though the surplus maximization mechanism is optimal (for social surplus), it is sometimes infeasible to run.

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Matching market n agents and m items Each agent i has a value v ij for item j V11 = 10,000$ V12 = 20,000$ V13 = 30,000$

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Matching market (cont.) The agents are unit-demand - each wants at most one item The items are unit-supply - each can be sold to at most one agent Agent values are drawn independently at random, v ij ∼ F ij.

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Item Pricing We start with only one agent, n = 1. We need to identify revenue optimal pricings I.e., identify a pricing p = (p1,..., pm) such that: when the agent buys the item that maximizesv j − p j → revenue is maximized.

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How we do that? Finding an upper bound on the revenue of an optimal pricings. if a pricing approximates this upper bound, it also approximates the optimal pricing.

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Finding an upper bound Fi rst, notice that instead of: Single agent, unit-demand preferences, with m items We create an environment of m (single-dimensional) agents, who each want their specific item, but with the constraint that at most one can be served.

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single-agent, m-item pricing V1V1 V4V4 V3V3 V5V5 V2V2 V i ~ F i

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single-item, m-agent auction V1 V4V3 V5 V2 V i ~ F i Only one can be served!

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Theorem For any product distribution F = F1 × · · ·× Fm, the expected revenue of the optimal single-agent, m-item pricing when the agent’s values for the items are drawn from F, is at most that of the optimal single-item, m-agent auction when the agents’ values for the item are drawn from F

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Expected revenue of the optimal single-item, m-agent auction V1 V4V3 V5 V2 Expected revenue of the optimal single-agent, m-item pricing ≥

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Pricing problem : seller can only post a price on each item. Auction problem: the competition between agents can drive the price up. Revenue in the (single-dimensional) auction environment is an upper bound on the revenue in the single buyer (multi-dimensional) pricing environment.

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Proof Any item pricing p can be converted into a single- item auction Ap such that the expected revenue from the item pricing is at most that of the auction. The auction Ap assigns the item to the agent j that maximizes v j − p j. For any fixed values of the other agents, v −j, this allocation rule is monotone in agent j’s value and therefore ex post incentive compatible. It is also deterministic. Our auction is deterministic, DSIC Mechanism So according to 2.18: there is a critical value j for agent j which is the infimum of values for which the agent wins the auction; the agent pays exactly this critical value on winning. Of course j ≥ p j2.18

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Reminder A direct, deterministic mechanism M is DSIC if and only if for all i: 2. (critical value) for 0 otherwise

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Proof (cont.) Also, notice that the allocation rule of the auction Ap is identical to the allocation rule of the pricing p: * For the pricing the agent chooses the item that maximizes v j − p j * For the auction the winner is selected to maximize v j −p j. * The revenue for the pricing is exactly the p j that corresponds to this j whereas in the auction it is τj which, is at least p j.

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Therefore, the auction Ap obtains at least revenue of the pricing p. Now we can say. Revenue in the (single-dimensional) auction environment is an upper bound on the revenue in the (multi- dimensional) pricing environment.upper bound

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Reminder: Theorem 4.7. There exists a threshold strategy such that the expected prize of the gambler is at least half the expected value of the maximum prize. Moreover, one such threshold strategy is the one where the probability that the gambler receives no prize is exactly 1/2. Moreover, this bound is invariant to the tie-breaking rule.

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Theorem 4.9. For any independent, single-item environment the second- price auction with a uniform ironed virtual reserve price is a 2-approximation to the optimal auction revenue. Theorem 4.10. For any independent, single-item environment a sequential posted pricing of uniform ironed virtual prices is a 2- approximation to the optimal auction revenue.

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7.2 Reduction: Unit-demand to Single- dimensional Preferences Goal: Show a reduction from multi-dimensional unit-demand preferences to single-dimensional preferences from the perspective of approximation. We assume independence of the agents values.

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Define a general unit-demand environment: n agents, m services. Agent i has value for service j. Indicator x with C() – cost function for the implicit constraint that each agent can only receive one service:

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7.2.1 Single-dimensional Analogy Definition 7.4 the representative environment for the n agent, m services unit demand environment given by F and c() is the single-dimensional environment given by F and c() with nm single-dimensional agents indexed by coordinates ij.

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7.2.2 Upper bound The restriction that only one representative of each unit demand agent can be served at once induces competition between representatives. Intuitively this competition should result in an increased revenue in the optimal mechanism for the representative environment over the original unit-demand environment.

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Theorem 7.5. For any independent, unit- demand environment, the optimal deterministic mechanism’s revenue is at most that of the optimal mechanism for the single- dimensional representative environment.

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single-item, nm-agent auction V i1 V i4 V i3 V i5 V i2 V i j ~ F ij V 1j V 2j V 3j V 4j V 5j

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Reminder Theorem 4.10. For any independent, single- item environment a sequential posted pricing of uniform ironed virtual prices is a 2- approximation to the optimal auction revenue. Sequential posted pricings, i.e., where the agents arrive in any order, approximate the optimal multi-agent single-item auction.

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7.2.3 Reduction We will now show a lower bound: In unit-demand pricing, item is allocated that In sequential posted pricing ties are broken in worst-case order to I.e. worst case is that the first agent arrives and pays the lowest price.

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Definition 7.6. A sequential posted pricing is an pricing of services (specialized) for each agent with the semantics that agents arrive in any order and take their favorite service that remains feasible. The revenue of such a pricing is given by the worst-case ordering.

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A sequential posted pricing is given by prices p with the price offered to agent i for item j. After the valuations are realized, the agents arrive in sequence and take their utility maximizing item that is still feasible.

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Example We assume worst case

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Theorem 7.7. The expected revenue of a sequential posted pricing for unit-demand environments is at least the expected revenue of the same pricing in the representative single-dimensional environment.

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Proof When comparing the two environments, in the representative environment the nm agents can arrive in any order. In the original environment, an agent arrives an considers the prices on services ordered by utility. The set of orders in the representative environment contains the set of orders in the original. For worst-case the representative is worse.

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Corollary 7.8. If a sequential posted pricing is approximately optimal in the representative (single-dimensional) environment it is approximately optimal in the original (unit- demand) environment.

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7.2.4 Instantiation we need to show that there are good sequential posted pricing mechanisms for single-dimensional environments. Here we will give such an instantiation for independent, regular, matching markets, i.e., where the services are items, and each item has only one unit of supply.

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The representative environment for matching markets: nm agents. Agent ij with value desires item j. For item j and original agent i, at most one representative ij can win. VSM is the optimal mechanism. - the probability that VSM serves ij..

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Definition 7.9. For the representative matching market environments, the simulation prices, p, satisfy

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Theorem 7.10. For regular distributions in the representative matching market environment, the sequential posted pricing with the simulation prices p is an 8-approximation to the revenue of the optimal mechanism. Lemma 7.11. For regular distributions in the representative matching market environment, the expected revenue of the optimal mechanism, VSM, is at most. Lemma 7.12. For regular distributions in the representative matching market environment, the expected revenue from the sequential posted pricing of the simulation prices is at least Upper Bound Lower Bound

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Upper Bound Consider an un-constrained mechanism that allocates to representative ij with probability at most Because if regularity the un-constrained mechanism posts price to each representative ij and the expected revenue is VSM is the optimal mechanism for the constrained environment and a valid solution to the un-constrained environment.

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Lower Bound. since prices increase with a lower selling probability. The expected revenue from agent ij is

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Lower Bound We need to show that the probability that it is feasible to offer service to representative ij is at least ¼. Representative ij is last and can be served if none of the other representatives has be served yet, and no one been served item j.

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Lower Bound Show that each of the events happens with probability at least 1/2, since the events are independent, the probability for both is ¼. Consider the event. The possibility that is a bad event happens with probability The of all the bad events is bounded by Therefore, probability that none of them happen is at least ½.

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Lottery Pricing and Randomized Mechanisms Optimal mechanism may not be a deterministic pricing of items. We now show another mechanism using pricing of items including pricing randomized outcome. Lotteries – Giving a price to randomized outcomes. A lottery is a probability distribution over outcomes. For instance, for the m = 2 item case, a lottery could assign either item 1 or item 2 with probability 1/2 each. A lottery pricing is then a set of lotteries and prices for each.

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For such a lottery pricing, the agent then chooses the lottery and price that give the highest utility for his given valuations for the items. Lottery pricings can give higher revenue than item pricings

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There are 2 items (m=2), 1 agent (n=1) The agent’s value for each item is distributed independently and uniformly from the interval [5,6]. Set a uniform price of 5.097 for each item. Agent is offered to buy an item at the price of 5.097 The agent buys the item he values the most, and also at least 5.097 Example V1 V2 Min price: 5.097

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Allocation rule without lottery

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We now provide another option to agent - buying at price 5.057 a lottery with probability ½ to get an item. Without the lottery option if the agent had average value bigger than 5.057 but no individual value over 5.097, the agent would buy nothing. Adding the lottery option V1V2 Min price: 5.097 or lottery (price: 5.057) Lottery – 5.057

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Allocation rule with lottery with price p’=5.057

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There are examples where the optimal lottery pricing obtains more revenue than the optimal single-item auction for the representative environment. Unlike representative environment that increased competition to obtain more revenue than that of the original unit-demand environment. It is not entirely correct when randomized mechanisms are allowed. Optimal lottery pricing

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Homework - Exercise 7.2 Prove Theorem 7.5: For any independent, unit-demand environment, the optimal deterministic mechanism’s revenue is at most that of the optimal mechanism for the single-dimensional representative environment.

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