Activity Rules: An ongoing saga …

1. Activity Rules: An ongoing saga… Adam Kirsch, Alex Kulesza, Loizos Michael Exchange / Activity Rules Group CS-286r Class Project Exchange Day Presentation

2. Agent’s view of Activity Rules • Provide the options of an agent at each step. • An agent has the right to choose how to satisfy the activity rules from the given choices. • An agent has the obligation to choose something, or else a default action is imposed. • Not a punishment, but rather an implicitly stated choice. • Activity rules should not discourage participation. CS-286r Class Project Exchange Day Presentation

3. Proxy’s view of Activity Rules • Advise agent on possible options for bidding. • Suggest options that optimize some criteria. • Point out consequences of violating activity rules. • Suggest how to minimize impact of consequences. • Advise on how likely is for the exchange to close. CS-286r Class Project Exchange Day Presentation

4. Exchange’s view of Activity Rules • Restrict an agent’s strategic behavior. • Drive exchange to an efficient outcome quickly. • Plus, an activity rule should be… CS-286r Class Project Exchange Day Presentation

5. List of Desiderata • Intuitive (not arbitrary, makes sense to people) • Easy to understand (use only few rules, or one!) • Easy to check (verify that bids are valid) • Easy to satisfy (truthful bidding always works) • Flexible (allows a change in the bid structure) • General (works for any bidding language) • Promotes fast convergence of prices • Hard to game (improve over truthful bidding) • Possibly hard to accommodate all these! But… CS-286r Class Project Exchange Day Presentation

6. Motivating Principles • …we are trying! • Abstract away from specific bidding languages, to see what the essence of a bid is. • Decompose activity rule provisions to smaller parts of the bid, so as it can be applied locally. • Do not change current way people think of bids. • Make agents commit to their choices / bids. CS-286r Class Project Exchange Day Presentation

7. Bids as Logical Operator Trees • Leaves are items and interior nodes are logical ops. • Some vertices have values s.t. there is exactly one value within each root to leaf path. • For upper/lower bounds, make each value a pair. CHOOSE(2) AND / $8-9 XOR / $3-5 OR A B C D E / $5-6 F / $2-4 CS-286r Class Project Exchange Day Presentation

8. The Interval Choose Operator • AND(S), OR(S), XOR(S), CHOOSE(k, S) are special cases of the interval choose operator IC(x, y, S). • Value for any subset S`S of items s.t. x|S`|y • AND(S)  IC(|S|, |S|, S) • OR(S)  IC(1, |S|, S) • XOR(S)  IC(1, 1, S) • CHOOSE(k, S)  IC(k, k, S) CS-286r Class Project Exchange Day Presentation

9. IC-Trees • Definition of an IC-tree: • Base Case (leaves): a single item. • Recursive Case (interior nodes): If S is a set of IC-trees, and 1 x,y |S|, then IC(x, y, S) is an IC-tree. CS-286r Class Project Exchange Day Presentation

10. IC(1,2) IC(1,1) IC(2,3) F A B C D E IC-Trees CS-286r Class Project Exchange Day Presentation

11. Satisfaction • An IC-tree T is satisfied by allocation A iff: • Base Case (leaves): T is a single item that belongs in A. • Recursive Case (interior nodes): T is of the form IC(x, y, S) and at least x elements of S are satisfied by A. CS-286r Class Project Exchange Day Presentation

12. Satisfaction IC(1,2) A = {C,D} IC(1,1) IC(2,3) F A B C D E CS-286r Class Project Exchange Day Presentation

13. IC(1,1) A B Bids - Adding Value to IC-Trees • Definition of an atomic bid: • An IC-Tree T paired with low/high bounds on the price an agent will pay for any satisfying allocation of T. $3-6 Abbreviated as: $3-6 CS-286r Class Project Exchange Day Presentation

14. Recursive Bids • Definition of a bid: • Base Case: An atomic bid. • Recursive Case: IC(x, y, S), where S is a set of bids and 1 x,y |S|. IC(1,2) IC(2,3) $3-3 $3-6 $1-8 $2-5 $1-4 CS-286r Class Project Exchange Day Presentation

15. The Activity Rule (for Buyers) • The activity rule is based on three objects: • The agent’s previous bid Bold • The previous round’s provisional allocation A • (based on the L/H outcome) • A low/high pair of activity rule price sets (PL , PH) • Derived from previous round’s linear prices CS-286r Class Project Exchange Day Presentation

16. The Activity Rule • A new bid B meets the activity rule iff: • The structure of B matches that of Bold (for now) • For every atomic bid b in B, the low (high) bound is at least (most) the corresponding bound in Bold • B is consistent with maximal commitment given A, PL, and PH. CS-286r Class Project Exchange Day Presentation

17. Maximal Commitment • Maximal Commitment: The agent demonstrates a continued interest in as many parts of its bid as possible (given its true valuation), up to its original stated interest. • Roughly speaking, a bid is consistent with maximal commitment if upper bounds are dropped wherever lower bounds are not raised: if an agent will not bid the current price now, it may never do so in the future. CS-286r Class Project Exchange Day Presentation

18. Full Commitment • Definition of full commitment: • Base Case: For atomic bid (T, L, H), either T is satisfied by A or L  mL = min PL(A`) over allocations A` satisfying T. • Recursive Case: For a bid of the form IC(x, y, S), at least y bids in S are fully committed. CS-286r Class Project Exchange Day Presentation

19. Full Commitment IC(1,2) IC(2,3) $3-3 $3-6 -> $4-6 $1-8 $2-5 $1-4 mL = $2 mL = $4 mL = $2 mL = $8 mL = $3 CS-286r Class Project Exchange Day Presentation

20. Maximal Commitment • If B demonstrates full commitment, then by definition it is maximally committed (has matched original interest). • If it does not, then by implication a lesser commitment is maximal given that agent’s true valuation. • Therefore, a default action is applied to keep B consistent with maximal commitment. CS-286r Class Project Exchange Day Presentation

21. The Default Action • If B does not reflect full commitment, the default action applies: • Base Case (T, L, H): Let mH = min PH (A`) over allocations A` satisfying T. Set H = min(H,max(L,mH)). • Recursive Case IC(x, y, S): Apply default action recursively to every bid in S not demonstrating full commitment. CS-286r Class Project Exchange Day Presentation

22. Agent Adjusts Bounds: IC(1,2) IC(2,3) $3-3 $3-6 $1-8 -> $2-8 $2-5 $1-4 -> $3-3 mL = $2 mL = $4 mL = $2 mL = $8 mL = $3 CS-286r Class Project Exchange Day Presentation

23. Full Commitment Bid Marking: IC(1,2) IC(2,3) $3-3 $3-6 $2-8 $2-5 $3-3 mL = $2 mL = $4 mL = $2 mL = $8 mL = $3 CS-286r Class Project Exchange Day Presentation

24. Default Action Propagation: IC(1,2) IC(2,3) $3-3 $3-6 $2-8 $2-5 $3-3 CS-286r Class Project Exchange Day Presentation

25. Default Action Base Case: IC(1,2) IC(2,3) $3-3 $3-6 -> $3-5 $2-8 $2-5 $3-3 mH = $5 mH = $9 CS-286r Class Project Exchange Day Presentation

26. Final Bid: IC(1,2) IC(2,3) $3-3 $3-5 $2-8 $2-5 $3-3 CS-286r Class Project Exchange Day Presentation

27. Closing Rule • The exchange closes if the L/H price vector has converged. • Convergence means that the L norm of the vector has changed by less than epsilon for each of the past N rounds. • Agents are not notified a priori of a final round, but epsilon and N are always available. CS-286r Class Project Exchange Day Presentation

28. Achieving the Desiderata Intuitive: • Agents must address every provisional allocation by • being content with it (default action / satisfied bid), • or modifying bid to beat “provisional price.” Easy to Understand: • A handful of definitions. • Based entirely on simple recursive structures. CS-286r Class Project Exchange Day Presentation

29. Justification Cont. Easy to Check: • Can compute values for m with depth-first search. • Then check compliance with simple recursion. Easy to Satisfy (Truthfully): • Raise some lower bounds. Implement default action on the rest. • If this is too demanding, we can introduce a relaxation. CS-286r Class Project Exchange Day Presentation

30. Justification Cont. Flexible: • Can easily add sub-bids anywhere. • But no deletions. • Apply rule retroactively as if new sub-bid was always there. • Possibly subject to gaming. • May break convergence. • Will not support this for the first implementation. CS-286r Class Project Exchange Day Presentation

31. Justification Cont. General: • Activity Rules are the same for all bidding languages that we have considered. • Plus interesting ones that we haven’t. Convergence: • Forces agents to address every provisional allocation (significant slack reduction). • Can make rules less demanding early on, if needed. CS-286r Class Project Exchange Day Presentation

32. Gaming: Dummy Business Plans • Naive dummy business plan strategy will not work. • The agent must always • declare interest in a partially satisfying allocation • declare non-interest in everything else. • More advanced dummy-bid strategies require more attention. CS-286r Class Project Exchange Day Presentation

33. Information for the Proxy • The rules admit many simple proxy computations. • Minimal lower bounds for bids to meet the activity rule. • Minimal number of changes. • Minimal change in sum of upper/lower bounds. • Possibly accommodate agent’s constraints? • Can imagine a pretty GUI with satisfied vertices lighting up as values are tweaked. CS-286r Class Project Exchange Day Presentation

34. Conclusions • Introduced simple rules for exchange with motivating philosophies and justifications. • Outlined the interactions between the rules and the groups that are affected by them. • Addressed some (all?) of the points of the rule-related confusion/concern that has arisen so far. • To do: Adding bids (changing the bid structure). CS-286r Class Project Exchange Day Presentation