在和扬尼斯教授交流后,我确定了 24 Spring 这 10 ECT 的 Projektarbejde i Datalogi 的具体方向为“不可分割物品的公平分配博弈” Fair Division with Indivisible Items

美美考完算法博弈论,从扬尼斯教授这拿了 12 分之后,我就得开搞 Projekt 了。

Objective

Gain familiarity with fairness notions and key results in fair division with indivisible items.

Fairness Notions

Denotation

Set NN of nn agents

Set MM of mm indivisible goods

For agent ii:

  • Valuation function vi:2MR0v_i:2^M\rightarrow\mathbb R_{\ge0}.

    vi()=0,vi(S)vi(T)STMv_i(\emptyset)=0,v_i(S)\le v_i(T)\forall S\subseteq T\subseteq M

    Assume valuation function are additive vi(S)=gSvi(g)v_i(S)=\bigcup_{g\in S}v_i(g) (My question: Does this imply single parameter environment? Why?)

Allocation instance I=(N,M,v)I=(N,M,v), v=(v1,,vn)v=(v_1,\cdots,v_n) is the vector of valuation functions.

Allocation A=(A1,,An)A=(A_1,\cdots,A_n) s. t. each agent ii receive the bundle AiMA_i\subseteq M, AiAj=ijA_i\cap A_j=\emptyset\forall i\neq j, iNAi=M\bigcup_{i\in N}A_i=M.

If iNAiM\bigcup_{i\in N}A_i\subsetneq M, the allocation is called partial.

Familiy of Envy-freeness

Envy-freeness

AA is EF if i,jN\forall i,j\in N, vi(Ai)vi(Aj)v_i(A_i)\ge v_i(A_j).

Deciding whether an instance admits EF is NP-complete.

EF is not always exist.

Envy-freeness up to one good

EF1, a relaxation of EF.

AA is EF1 if i,jN\forall i,j\in N, vi(Ai)vi(Aj{g})v_i(A_i)\ge v_i(A_j\diagdown\{g\}) for some gAjg\in A_j.

Very weak notion: gg maybe high-valued.

My difinition: AA is EF1 if i,jN,vi(Ai)vi(Ai{g})\forall i,j\in N,v_i(A_i)\ge v_i(A_i\diagdown\{g\}) for g=argmaxtAjvi(t)g=\arg\max_{t\in A_j}v_i(t)

Computing EF1

Round Robin Algoithm

Envy-Cycle elimination

EF1 and Pareto Optimality

MNW allocation: an allocation AA is MNW if

  1. It maximizes number of agents with positive value
  2. It maximizes the Nash welfare (the product of valuation) for agents with positive value.

MNW\rightarrowEF1+PO

❓Can EF1+PO be computed in P time?

Envy-freeness up to any good

EFX, a relaxation of EF stricter than EF1.

AA is EFX if i,jN\forall i,j\in N, vi(Ai)vi(Aj{g})v_i(A_i)\ge v_i(A_j\diagdown\{g\}) for any gAjg\in A_j.

❓Open problem: The existence of EFX? (n4n\ge4? Unrestricted additive valuations?)

My difinition: AA is EFX if i,jN,vi(Ai)vi(Ai{g})\forall i,j\in N,v_i(A_i)\ge v_i(A_i\diagdown\{g\}) for g=argmintAjvi(t)g=\arg\min_{t\in A_j}v_i(t)

Approximately EFX

α\alpha-EFX: α[0,1]\alpha\in[0,1] if i,jN\forall i,j\in N, vi(Ai)α×vi(Aj{g})v_i(A_i)\ge\alpha\times v_i(A_j\diagdown\{g\}) for any gAjg\in A_j.

12\frac{1}{2}-EFX always exists for sub-additive valuation functions.

Envy-Cycle Elimination algorithm computes 121\over2-EFX allocations.

❓What is the best possible α\alpha for α\alpha-EFX allocation?

Loose: must allocate all goods\rightarrowallocate part of goods.

❓Is it possible to achieve an exact EFX allocation by donating a sub-linear number of goods?

Other types of Envy-freeness

Envy-freeness up to one less-preferred good. EFL, between EF1 and EFX.

  • EFL always exists, can be computed using a variant of Envy-Cycle Elimination.

Envy-freeness up to a random good. EFR

  • Randomized version.
  • 0.730.73-EFR allocation can be computed in polynomial time.

❓Do EFR allocations always exist?

Epistemic Fairness

  • Agents don’t have complete knowledge about allcation.

Familiy of Proportionality

Proportionality

AA is PROP if iN\forall i\in N, vi(Ai)vi(M)nv_i(A_i)\ge\frac{v_i(M)}{n}.

Obviously EF\rightarrowPROP, PROP↛\not\rightarrowEF.

Deciding whether an instance admits PROP is also NP-complete.

Prop1

Propotionality up to one good

Prop1+PO always exists, can be computed in P time.

PropX

Propotionality up to any good

Can not always be guaranteed.

PropM

Propotionality up to maximin good

Between Prop1 and PropX

The value of a maximin good for ii is di(A)=maxjimingAjvi(g)d_i(A)=\max_{j\neq i}\min_{g\in A_j}v_i(g)

AA is called PropM if vi(Ai)+di(A)vi(M)nv_i(A_i)+d_i(A)\ge\frac{v_i(M)}{n},

PropM always exists, can be computed in P time.

Maximin Share Fairness

MMS, a relaxation of PROP, is trying to give each agent ii bundles of value at least as her maximin share μin(M)\mu_i^n(M).

Maximin share is the worst value agent can guarantee for herself by partitioning MM into nn disjoint bundles and keeping the worst of them.

Let An(M)\mathcal A_n(M) be te collection of possible allocations of the goods in MM to nn agents.

AA is MMS if iN\forall i\in N:

vi(Ai)μin(M)=maxBAn(M)minSBvi(S)v_i(A_i)\ge\mu_i^n(M)=\max_{B\in\mathcal A_n(M)}\min_{S\in B}v_i(S)

Computing MMS even Maximin share of an agent is NP-hard.

MMS don’t always exist when there are more than 2 agents.

Approximately MMS

α\alpha-MMS: iN\forall i\in N, vi(Ai)α×μin(M)v_i(A_i)\ge\alpha\times\mu_i^n(M)

❓Is it possible to improve upon the bound of 34+112n\frac{3}{4}+\frac{1}{12n} for additive valuations? Is there a stronger inapproximability bound than 394039\over40?

❓Are there other class of structured valuations for which MMS is guaranteed to exist?

PMMS

Pairwise maximin share fairness

i,jN,vi(Ai)maxAiAj=AiAjmin{vi(Ai),vi(Aj)}\forall i,j\in N,v_i(A_i)\ge\max_{A_i'\cup A_j'=A_i\cup A_j}\min\{v_i(A_i'),v_i(A_j')\}

α\alpha-PMMS

❓Do PMMS allocations always exist?

GMMS

Groupwise maximin share fairness

α\alpha-GMMS: if iN\forall i\in N, vi(Ai)αGMMSiv_i(A_i)\ge\alpha\cdot\text{GMMS}_i

GMMSi=maxSN:iSμiS(jSAj)\text{GMMS}_i=\max_{S\subseteq N:i\in S}\mu_i^{|S|}(\bigcup_{j\in S}A_j)

Best known: 474\over7-GMMS

❓What’s the best possibe α\alpha for α\alpha-GMMS?

Efficiency and Incentives

Fair and Pareto Optimal Allocations

Approximating the Nash Welfare

Price of Fairness

PoF=supIminAF(O)OPT(I)SW(A)=maxDDivisionSW(D)maxSFair DivisionSW(S)\text{PoF}=\sup_I\min_{A\in F(O)}\frac{OPT(I)}{SW(A)}\\ =\frac{\max_{D\in\text{Division}}SW(D)}{\max_{S\in\text{Fair Division}}SW(S)}

Fair Divsion with Strategic Agents