1. Designing Incentives for Boolean Games Ulle Endriss, Sarit Kraus Jerome Lang, Michael Wooldridge presented by Boris Trayvas

2. Outline Introduction to a new problem No Algorithms How can we use Incentives in Games?

3. Taxes Incentivize socially desired behaviors: taxes on import to encourage the local market taxes on transportation to encourage public transportation, etc.

4. Propositional Logic – a quick reminder B = {T,F} truth values Φ = {p,q,….} variables (fixed, finite, nonempty) L set of all WFF over Φ v: Φ  B valuation V set of all valuations over Φ

5. What are Boolean Games? A Boolean game with n participants is a (2n+3)- tuple. G is an n participants Boolean game, if G= and: Ag= {1,…,n} agents Φ = {p,q,….} variables c: ΦX B  R⁺ ϒ1,…, ϒn ϵ L ϕ1,…., ϕn is a partition of Φ i.e., (Φ= U ϕi) & (i≠j  ϕi ∩ ϕj = {})

6. What are Boolean Games? Choices: vi : ϕi  B a legal choice of truth values. Vi is the set of all choices for agent i. Outcome: An outcome (v1,…,vn) ϵ (V1 X…X Vn) is a collection of choices, one for each agent.

7. What are Boolean Games? succ(v1,…,vn) = { i ϵ Ag | ϒ i ⊨ (v1,…,vn) } c: Φ X B  R⁺ c(p,b) is the cost of assigning variable p to value b. C0 ≡ 0

8. Goal Each agent i has two goals: Main goal: satisfy ϒi Secondary goal: minimize the cost of the assignments.

9. Example G= Ag= {1,2} Φ = {p,q} c(p,T)=c(q,T)=5 c(p,F)=c(q,F)=2 ϕ 1={p} ϒ1 = p ϕ 2={q} ϒ2 = ~p Possible moves: v1(p) = T v2(q) = F (cheaper option)

10. Taxation Schemes 1 Let G be a Boolean game, and let ϒ ϵ L (WFF over Φ) Assume there is a principal that wishes to achieve ϒ but can not directly interfere in the agents choices. His only way to influence the game is by posing taxes on the players. Costs are internal, Taxes are external. How will we define T?

11. Taxation Schemes 2 T: Φ X B  R⁺ with the meaning: T(p,b) = the tax imposed on the owner of p for giving it the truth value b. For a valuation vi, T(vi) = ∑ pϵϕi T(p,vi(p)) We only consider taxation schemes that can be represented in polynomial (relative to |G|) space.

12. Utilities and Preferences We will extend cost and taxation to partial valuations: C(vi) = ∑ pϵϕi C(p,vi(p)) T(vi) = ∑ pϵϕi T(p,vi(p)) Mi = max viϵVi (C(vi) + T(Vi)) Mi is the most expensive choice.

13. Utilities and Preferences 2 Utility of agent i: Ui(v1,…,vn) = 1+Mi-(C(vi) + T(vi)) if (v1,…,vn) satisfies ϒ -(C(vi) + T(vi)) otherwise Properties of this definition: -Mi≤ Ui ≤ 1+Mi An agent will always prefer to satisfy ϒ An agent will prefer the cheapest option

14. Solution Concepts We will focus on Nash Equilibrium as solutions for a game. (v1,…,vn) is a Nash Equilibrium if for all i ϵ Ag there is no wi ϵ Vi s.t. Ui(v1,…,wi,…,vn) > Ui(v1,…,vi,…vn) NE(G,T) is the set of all Nash Equilibrium for a Boolean game G and taxation scheme T.

15. Proposition 1 Suppose that (v1,…,vi,…,vn) ϵ NE(G,T) and that (v1,…,vi,…,vn) doesn’t satisfy ϒi Then vi = min(Ci(vi) + Ti(vi))

16. Incentive Design Given a Boolean game G, The Principle wants to provide incentives for the Agents of G to reach a certain collective outcome, denoted ϒ. The Principle is external to G, but can impose a taxation scheme T on G.

17. Weak Implementation A taxation scheme T will be named a weak implementation if {vϵNE(G,T) | v satisfies ϒ} is not empty. That is, if there is at least one Nash equilibrium which satisfies The Principals goals.

18. Example of WI Ag = {1,2} Φ={p,q} ϕ1 = {p} ϕ2 = {q} ϒ1 = p ϒ2 = ~p & ~q ϒ = p & q C(p,T)=C(p,F)= C(q,F) =0 C(q,T) = 1 1.With T = 0, there is a single NE: p=T,q=F, but obviously ϒ is not satisfied. 2.Define T’:T’(q,F) = 10, T’(p,?)=T’(q,T)=0 NE(G,T’) = {(p=T,q=T)} and thus ϒ is satisfied.

19. Sufficient Condition for WI What is a sufficient condition for the existence of a weak implementation? ϒ is satisfiable ? ϒ is a tautology?

20. (un)Sufficient Condition for WI 1 ϒ being satisfiable is not enough to ensure the existence of a WI. Consider the following game G: Ag = {1} Φ = ϕ1 = {p} ϒ1 = p C = C0 ϒ = ~p

21. (un)Sufficient Condition for WI 2 ϒ being a tautology is not enough to ensure the existence of a WI. Consider the following game G: Ag = {1,2} Φ = {p,q} ϕ1 = {p} ϕ2 = {q} ϒ1 = (p↔q) ϒ2 = ~(p↔q) C = C0 ϒ = T Finding a taxation scheme that ensures at least one NE is called the STABILISATION problem.

22. Sufficient Condition for WI 3 ϒ’ := ϒ & (ϒ1 & ϒ2 & … & ϒn ) ϒ’ satisfiable → there exists a WI Intuition: Taking a valuation v that satisfies ϒ’, all we need to do is set the taxes high enough so that choosing v is the cheapest option.

23. Strong Implementation Sometimes knowing that there exists a NE that satisfies ϒ is not enough, what if we want ALL NE to satisfy ϒ? SI(G, ϒ) := the set of taxation schemes T over G for which: (G,T) has at least one NE All NE of (G,T) satisfy ϒ

24. Relation between WI and SI It is easy to see, that every SI is also a WI. Not every WI is a SI. To show no SI exists, it Is enough to show there is no WI.

25. Simplifying Things: For each game (G,T), there exists (G’,T’) with cost function C0 s.t. NE(G,T)=NE(G’,T’) Proof: define T’(p,b) = T(p,b) + c(p,b)  utility Ui doesn’t change  NE(G,T)=NE(G’,T’) A taxation scheme T will be called positive if T(p,F) = 0 It is possible to show that it is also sufficient to only consider positive taxations.

26. Social Welfare Usually, there will not be exactly one taxation scheme T satisfying ϒ. Not all of the schemes are equally diserable by society. This considerations are secondary to satisfying ϒ. We will mention the different approaches, but will not discuss them deeply.

27. Utilitarian Social Welfare usw(v1,…,vn) = ∑ iϵAg Ui(v1,…,vn) It is possible to show that maximizing USW and satisfying ϒ at the same time is sometimes impossible. Doesn’t ensure equality. Indeed, sometimes the best thing to do, is ‘give’ a specific agent all the utility, leaving the others with none.

28. Egalitarian Social Welfare esw(v1,…,vn) = min{ui(v1,…,vn)|iϵAg} The best taxation scheme, is the one where the poorest agent is the richest. Avoids the problems mentioned about USW.

29. Minimizing the Total Tax Burden maximizing social welfare = minimizing tax burden Of course, we still need to satisfy ϒ. We are looking for T such that tb(v1,…,vn) = ∑ iϵAg T(vi) is minimal.

30. Taxation And Equity As we have noticed, Social Welfare is not always fare. Other ways to compare between tax schemes include: Minimizing the Difference in Taxes Horizontal Equity: Same as the previous, but we divide our agents into two classes – those who succeeded and those who didn’t.

31. Future Work Negative Taxes (k-implementation) When can ϒ be implemented in G? Using Taxation schemes outside of Boolean Games. Why Nash Equilibrium? Why not something else? (Dominant strategy for example).

32. Questions?