Automated Design of Multistage Mechanisms Tuomas Sandholm (Carnegie Mellon) Vincent Conitzer (Carnegie Mellon) Craig Boutilier (Toronto)

Mechanism design An outcome must be chosen from a set of outcomes Every agent has preferences over the outcomes, represented by a type –We only know priors over agents’ types Each agent reports its type to the mechanism, mechanism chooses outcome –mechanism = function from type vectors to outcomes –auction rules, voting rules, … But: agents will lie if this is to their benefit! Solution: mechanism should be designed so that agents have no incentive to lie –justified by revelation principle Some general mechanisms exist (e.g. VCG) but they are not always applicable/optimal

Automated mechanism design [Conitzer & Sandholm UAI-02] Idea: design optimal mechanism specifically for setting at hand, as solution to an optimization problem Generates optimal mechanisms in settings where existing general mechanisms do not apply/are suboptimal If mechanism is allowed to be randomized, can be done in polynomial time using linear programming (if number of agents is small)

Small example: Divorce arbitration Outcomes: Each agent is of high type with probability 0.2 and of low type with probability 0.8 –Preferences of high type: u(get the painting) = 100 u(other gets the painting) = 0 u(museum) = 40 u(get the pieces) = -9 u(other gets the pieces) = -10 –Preferences of low type: u(get the painting) = 2 u(other gets the painting) = 0 u(museum) = 1.5 u(get the pieces) = -9 u(other gets the pieces) = -10

Optimal randomized, dominant strategies, single-stage mechanism for maximizing sum of divorcees’ utilities high low high

The elicitation problem In general, every agent must report its whole type This may be impractical in larger examples –In a combinatorial auction, each agent has values for exponentially many bundles –Computing one’s type may be costly –Privacy loss For most mechanisms, this is not necessary –E.g. second-price auction only requires us to find the winner and the second-highest bidder’s valuation Multistage mechanisms query the agents sequentially for aspects of their type, until they have determined enough information –E.g. an English auction Can we automatically design multistage mechanism?

A multistage mechanism corresponding to the single-stage mechanism low high low high

Saving some queries low high low high with probability.4, exit early with

Asking the husband first low high low high

Saving some queries (more this time) low high low high with probability.51, exit early with.92.08

Changing the underlying mechanism For the given optimal single-stage mechanism, we can save more wife-queries than husband-queries Suppose husband-queries are more expensive We can change the underlying single-stage mechanism to switch the roles of the wife and husband (still optimal by symmetry) If we are willing to settle for (welfare) suboptimality to save more queries, we can change the underlying single-stage mechanism even further

Fixed single-stage mechanism, fixed elicitation tree As we saw: If all of a node’s descendants have at least a given amount of probability on a given outcome, then we can propagate this probability up Theorem. Suppose both the single-stage mechanism and the elicitation tree (query order) are fixed. If we propagate probabilities up as much as possible, we get the maximum possible savings in terms of number of queries.

What if the tree is not fixed? Construct the tree first, then we can propagate up as before Observation: The exit probability at a node does not depend on the structure of the tree after it A greedy approach to asking queries: next query = query maximizing the probability of exiting right after it –Time complexity: O(|Q|*|A|*|O|*|Θ|) Proposition. In various (small) examples, the greedy approach can save only an arbitrarily small fraction of the queries saved with the optimal tree

Finding the optimal tree using dynamic programming After receiving certain answers to certain questions, we are in some information state Dynamic program computes the (minimum) expected number of queries needed from every state (given that we have not exited early) Time complexity: O(|Q|*|A|*|O|*|Θ|*2 |Θ| )

What if underlying single- stage mechanism is not fixed (but elicitation tree is)? Approach: design single-stage mechanism taking eventual query savings into account Single-stage mechanism is designed using linear programming techniques So, can we express query savings linearly? Yes: For every vertex v in the tree, let –c(v) be the cost of the query at v –P(v) be the probability that v is on the elicitation path –e(v) the probability of exiting early at or before v given that v is on the elicitation path Then, the query savings is Σ v c(v)P(v)e(v) All of these are constant except e(v) = Σ o min θ  v p(θ, o)

What if nothing is fixed? Could apply previous approach to all possible trees (inefficient) No other techniques here yet…

Auction example One item, two bidders with values uniformly drawn from {0, 1, 2, 3} Objective: maximize revenue Optimal single-stage mechanism generated: (compare: Myerson auction)

Multistage version of same mechanism Using the dynamic programming approach for determining the optimal tree, we get:

Changing the underlying single-stage mechanism Using tree generated by dynamic program, we optimized the underlying mechanism for cost of per query Same expected revenue, fewer queries

Changing the underlying single-stage mechanism Same tree, but with a cost of 0.5 per query: Lower expected revenue, fewer queries

Beyond dominant-strategies single-stage mechanisms So far, we have focused on dominant strategies incentive compatibility for the single-stage mechanism –Any corresponding multistage mechanism is ex-post incentive compatible Weaker notion: Bayes-Nash equilibrium (BNE) –Truth-telling optimal if each agent’s only information about others’ types is the prior (and others tell the truth) –Multistage mechanisms may break incentive compatibility by revealing information Proposition. There exist settings where –the optimal single-stage BNE mechanism is unique –the unique optimal tree for this mechanism is not incentive compatible –there is a tree that randomizes over the next query asked that is BNE incentive compatible and obtains almost the same query savings as the optimal tree, more than any other tree

Conclusions For dominant-strategies mechanisms, we showed how to: –turn a single-stage mechanism into its optimal multistage version when the tree is given (propagate probability up) –turn a single-stage mechanism into a multistage version when the tree is not given greedy approach (suboptimal, but fast) dynamic programming approach (optimal, but inefficient) –generate the optimal multistage mechanism when the tree is given but the underlying single-stage mechanism is not BNE mechanisms seem harder (need randomization over queries) Thank you for your attention!