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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 \(N\) of \(n\) agents

Set \(M\) of \(m\) indivisible goods

For agent \(i\):

  • Valuation function \(v_i:2^M\rightarrow\mathbb R_{\ge0}\).

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

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

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

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

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

Familiy of Envy-freeness

Envy-freeness

\(A\) is EF if \(\forall i,j\in N\), \(v_i(A_i)\ge v_i(A_j)\).

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.

\(A\) is EF1 if \(\forall i,j\in N\), \(v_i(A_i)\ge v_i(A_j\diagdown\{g\})\) for some \(g\in A_j\).

Very weak notion: \(g\) maybe high-valued.

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

Computing EF1

Round Robin Algoithm

Envy-Cycle elimination

EF1 and Pareto Optimality

MNW allocation: an allocation \(A\) 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\(\rightarrow\)EF1+PO

❓Can EF1+PO be computed in P time?

Envy-freeness up to any good

EFX, a relaxation of EF stricter than EF1.

\(A\) is EFX if \(\forall i,j\in N\), \(v_i(A_i)\ge v_i(A_j\diagdown\{g\})\) for any \(g\in A_j\).

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

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

Approximately EFX

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

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

Envy-Cycle Elimination algorithm computes \(1\over2\)-EFX allocations.

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

Loose: must allocate all goods\(\rightarrow\)allocate 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.
  • \(0.73\)-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

\(A\) is PROP if \(\forall i\in N\), \(v_i(A_i)\ge\frac{v_i(M)}{n}\).

Obviously EF\(\rightarrow\)PROP, PROP\(\not\rightarrow\)EF.

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 \(i\) is \(d_i(A)=\max_{j\neq i}\min_{g\in A_j}v_i(g)\)

\(A\) is called PropM if \(v_i(A_i)+d_i(A)\ge\frac{v_i(M)}{n}\),

PropM always exists, can be computed in P time.

Maximin Share Fairness

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

Maximin share is the worst value agent can guarantee for herself by partitioning \(M\) into \(n\) disjoint bundles and keeping the worst of them.

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

\(A\) is MMS if \(\forall i\in N\): $$ v_i(A_i)\ge\mu_i^n(M)=\max_{B\in\mathcal A_n(M)}\min_{S\in B}v_i(S) $$ 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

\(\alpha\)-MMS: \(\forall i\in N\), \(v_i(A_i)\ge\alpha\times\mu_i^n(M)\)

❓Is it possible to improve upon the bound of \(\frac{3}{4}+\frac{1}{12n}\) for additive valuations? Is there a stronger inapproximability bound than \(39\over40\)?

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

PMMS

Pairwise maximin share fairness $$ \forall i,j\in N,v_i(A_i)\ge\max_{A_i'\cup A_j'=A_i\cup A_j}\min{v_i(A_i'),v_i(A_j')} $$ \(\alpha\)-PMMS

❓Do PMMS allocations always exist?

GMMS

Groupwise maximin share fairness

\(\alpha\)-GMMS: if \(\forall i\in N\), \(v_i(A_i)\ge\alpha\cdot\text{GMMS}_i\) $$ \text{GMMS}i=\max{S\subseteq N:i\in S}\mu_i^{|S|}(\bigcup_{j\in S}A_j) $$ Best known: \(4\over7\)-GMMS

❓What's the best possibe \(\alpha\) for \(\alpha\)-GMMS?

Efficiency and Incentives

Fair and Pareto Optimal Allocations

Approximating the Nash Welfare

Price of Fairness $$ \text{PoF}=\sup_I\min_{A\in F(O)}\frac{OPT(I)}{SW(A)}\ =\frac{\max_{D\in\text{Division}}SW(D)}{\max_{S\in\text{Fair Division}}SW(S)} $$ Fair Divsion with Strategic Agents