Answer: No online, exactly prefix-welfare-maximizing DSIC rule exists for the stated class of costs.
Theorem 1 — class: other
PDF text-layer excerpt — equations and symbol order may be approximateGiven limited misreports and non-increasing marginal valuations ˆvi of agents i ∈I, any mechanism with 2The assumption that marginal valuations are non-increasing is important here.
statement extracted from the paper’s text layer; consult the paper for the exact wording
For a finite horizon, suppose a finite-support distribution of EV types \((a,d,v)\), with rational masses, arrives over time. Given known aggregate cost curves \(C_t(q)=\int_0^q c(t,z)\,dz\), can one compute an online feasible allocation and payment rule that assigns each type’s mass among integral per-EV charging schedules, maximizes welfare for the types observed so far, and makes truthful reporting a dominant strategy for each EV?
Time consists of \(m\) slots. A society is given by its support: the list of \(\tau\) types with positive mass, each mass a binary rational; \(\tau\) is the support size, not the number of possible types. Masses represent an atomless population and need not sum to one. Only arriving support entries are revealed online.
A type is \(\theta=(a,d,v)\), where \(1\leq a\leq d\leq m\) and \(v\in\mathbb{Q}_{\geq0}^{m}\) has non-increasing coordinates. An integral per-EV schedule is \(s\in\{0,1\}^{m}\), zero outside \([a,d]\); its value is \(V_\theta(s)=\sum_{j=1}^{\sum_t s_t}v_j\). Electricity supply is unlimited.
Known costs are \(C_t(q)=\int_0^q c(t,z)\,dz\), with nonnegative piecewise-constant marginal costs specified by rational breakpoints and heights, including a final unbounded interval; monotonicity is not assumed. Let \(K\) count all pieces and \(L\) be the total binary input length.
For each observed type, allocate its mass among integral schedules. If \(x_{\theta,s}\) is schedule mass, require \(x_{\theta,s}\geq0\) and \(\sum_s x_{\theta,s}=\mu_\theta\); welfare is \(\sum_{\theta,s}x_{\theta,s}V_\theta(s)-\sum_t C_t(\sum_{\theta,s}x_{\theta,s}s_t)\).
At each slot, maximize this welfare for all reports observed so far, subject to each EV's implemented charging prefix. Implement the current slot; future assignments may be revised, but delivered energy cannot be cancelled. Decisions cannot use future arrivals, and any welfare-maximizing tie may be chosen.
Schedule shares define a lottery for a representative infinitesimal EV. Its utility is expected true value minus expected payment, assessed at reported departure using only information then available. DSIC means truthful reporting maximizes this utility against every stream of others' reports; admissible deviations satisfy \(a\leq\hat a\leq\hat d\leq d\) and may replace \(v\) by any admissible valuation vector. An individual's deviation does not change aggregate masses.
Does such an online, exactly prefix-welfare-maximizing DSIC rule exist? If it exists, can its allocations and payments be computed in polynomial time in \(m,\tau,K,L\), or is their computation NP-hard?
For discrete EVs, report-independent prices that cannot decrease for future slots, terminal utility-maximizing quantities, and compatible feasible schedules yield DSIC; this does not establish exact welfare optimality ([Hayakawa et al., 2015](https://www.ijcai.org/Proceedings/15/Papers/370.pdf)).
Price-based DSIC mechanisms also extend to discrete multi-unit demand with uncertain procurement costs ([Hayakawa et al., 2018](https://www.ifaamas.org/Proceedings/aamas2018/pdfs/p309.pdf)); no result settling exactly the atomless problem above was located.
No such rule exists for the allowed class of costs. A two-slot instance forces allocations that violate DSIC, regardless of payments or tie-breaking. The conditional computational question therefore does not arise: the impossibility persists with unlimited computation.
Let \(m=2\). The known marginal costs are
\(c(1,z)=5\) for all \(z\geq 0\), and
\(c(2,z)=10\) for \(0\leq z<2\), while \(c(2,z)=0\) for \(z\geq 2\).
Thus \(C_1(q)=5q\) and \(C_2(q)=10\min\{q,2\}\). These are admissible nonnegative piecewise-constant marginal costs with a final unbounded interval.
The report stream contains the following three positive-mass types:
| Type | Arrival | Departure | Valuation vector | Mass |
|---|---:|---:|---|---:|
| \(A\) | \(1\) | \(2\) | \((10,1)\) | \(1\) |
| \(B\) | \(1\) | \(2\) | \((4,3)\) | \(1\) |
| \(H\) | \(2\) | \(2\) | \((100,0)\) | \(3\) |
Every valuation vector is nonnegative and non-increasing. All data are integers, hence binary rationals. The support size is \(\tau=3\), and the costs have \(K=3\) pieces.
Types \(A\) and \(B\) arrive at slot \(1\); type \(H\) is revealed only at slot \(2\).
At slot \(1\), only types \(A\) and \(B\) have been observed. Their combined mass is \(2\), so every feasible plan has second-slot load \(q_2\leq 2\). Consequently, throughout this optimization,
\(C_2(q_2)=10q_2\).
The objective therefore separates into schedule welfare contributions. Writing schedules as \((s_1,s_2)\), these contributions per unit of type mass are:
| Schedule | Type \(A\) | Type \(B\) |
|---|---:|---:|
| \((0,0)\) | \(0\) | \(0\) |
| \((1,0)\) | \(10-5=5\) | \(4-5=-1\) |
| \((0,1)\) | \(10-10=0\) | \(4-10=-6\) |
| \((1,1)\) | \(11-5-10=-4\) | \(7-5-10=-8\) |
Because each type’s schedule masses sum to its given mass, its contribution is a weighted average of the corresponding entries. Each column has a unique maximum. Hence every welfare maximizer assigns all type \(A\) mass to \((1,0)\) and all type \(B\) mass to \((0,0)\).
Implementing slot \(1\) therefore forces:
There is no relevant tie-breaking freedom.
At slot \(2\), type \(H\) arrives. Let \(y_A,y_B,y_H\) denote the masses charged in that slot. The implemented prefixes require
\(0\leq y_A\leq 1,\quad 0\leq y_B\leq 1,\quad 0\leq y_H\leq 3\).
Type \(A\) has already received its first unit, so its second-slot marginal value is \(1\). Types \(B\) and \(H\) would receive their first units. Including the fixed first-slot welfare, the objective is
\(W(y_A,y_B,y_H)=5+y_A+4y_B+100y_H-10\min\{y_A+y_B+y_H,2\}\).
First, every maximizer must have \(y_H=3\). Indeed, if \(y_H<3\), increasing it by any feasible positive amount \(\delta\) increases value by \(100\delta\), while increasing cost by at most \(10\delta\). Welfare increases by at least \(90\delta>0\).
Once \(y_H=3\), the second-slot cost is constantly \(20\), regardless of \(y_A\) and \(y_B\). Their strictly positive objective coefficients therefore force \(y_A=y_B=1\).
Thus every exactly prefix-welfare-maximizing rule produces final schedules
\(s_A=(1,1),\qquad s_B=(0,1)\).
In particular, report \(A\) receives two units, while report \(B\) receives one unit.
Fix the aggregate report stream above. Consider a representative infinitesimal EV with availability window \([1,2]\). By assumption, changing its report does not change aggregate masses. Both reports \(A\) and \(B\) belong to the positive-mass support, and their schedule-share lotteries are forced by the preceding uniqueness arguments.
Let \(p_A\) and \(p_B\) be this EV’s expected payments when reporting \(A\) and \(B\), respectively, in this fixed aggregate environment. These expectations may include any mechanism randomness. Both reports have departure \(2\).
For a true type \(A\), truthful reporting gives value \(11\), whereas reporting \(B\) gives one unit and value \(10\). DSIC requires
\(11-p_A\geq 10-p_B\),
hence
\(p_A-p_B\leq 1\).
For a true type \(B\), truthful reporting gives value \(4\), whereas reporting \(A\) gives two units and value \(7\). DSIC requires
\(4-p_B\geq 7-p_A\),
hence
\(p_A-p_B\geq 3\).
These inequalities are incompatible. Both deviations are admissible: they preserve arrival and departure and replace the valuation vector by another admissible vector.
Exact online prefix welfare maximization uniquely determines allocations in this instance that no expected payments can make DSIC. The contradiction even allows payments to use the entire report stream through departure.
Therefore the requested rule does not exist under the stated hypotheses.
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The prover correctly establishes that no such rule exists for the permitted class of costs. The counterexample forces allocations that violate a necessary DSIC condition, independently of payments and tie-breaking. No major or minor issues were found.
All three types satisfy the required arrival and departure restrictions. Their valuation vectors are nonnegative and non-increasing, and their masses are positive binary rationals.
The marginal costs are nonnegative and piecewise constant, with rational data and final unbounded intervals. Integrating them gives exactly
\[
C_1(q)=5q,\qquad C_2(q)=10\min\{q,2\}.
\]
The decrease in second-slot marginal cost is permitted because the task explicitly does not assume monotonicity. The stated counts \(\tau=3\) and \(K=3\) are correct.
Only types \(A\) and \(B\) are observed at slot \(1\). Each EV can receive at most one unit in slot \(2\), so their combined mass of \(2\) implies \(q_2\leq2\). Consequently, second-slot costs are linear throughout this optimization.
Every entry in the schedule-welfare table is correct. In particular, the two-unit schedule has value \(11\) for \(A\) and \(7\) for \(B\), and total cost \(15\).
With both costs linear on the feasible domain and supply unlimited, the objective separates across schedule masses. Type \(A\) uniquely maximizes its contribution with schedule \((1,0)\), while type \(B\) uniquely maximizes its contribution with \((0,0)\). Any positive mass on another schedule strictly reduces welfare.
Thus the implemented first-slot prefixes are forced. This argument uses no information about the future arrival of \(H\).
The bounds on \(y_A,y_B,y_H\) correctly describe all feasible second-slot allocations consistent with those implemented prefixes.
The objective
\[
W=5+y_A+4y_B+100y_H-10\min\{y_A+y_B+y_H,2\}
\]
is correct. The constant \(5\) is the first-slot value of \(A\), minus its first-slot cost. The remaining coefficients correctly represent incremental values.
For any feasible increase \(\delta>0\) in \(y_H\), the cost increase is at most \(10\delta\), whereas the value increase is \(100\delta\). Hence every maximizer has \(y_H=3\).
Then total second-slot load is at least \(3\), making second-slot cost identically \(20\). The strictly positive coefficients of \(y_A\) and \(y_B\) force both to equal \(1\).
The resulting schedules \((1,1)\) for \(A\) and \((0,1)\) for \(B\) are therefore forced, including at the level of representative schedule-share lotteries.
The comparison uses the same fixed aggregate environment for both reports. This is justified by the task’s explicit assumption that an individual deviation does not change aggregate masses. Both compared reports belong to the positive-mass support, so their representative lotteries are specified and, here, deterministic.
The expected payments \(p_A,p_B\) describe the same individual’s report-dependent outcomes in that environment. They may include mechanism randomness. Both reports have departure \(2\), so the information available at payment assessment is the same.
For true type \(A\), the values of the two outcomes are \(11\) and \(10\). Its DSIC inequality correctly implies
\[
p_A-p_B\leq1.
\]
For true type \(B\), those values are \(7\) and \(4\), respectively. Its DSIC inequality correctly implies
\[
p_A-p_B\geq3.
\]
These conditions are incompatible. Both deviations preserve the availability window and use admissible valuation vectors. The contradiction therefore applies to the required deviation domain.
One admissible instance suffices to disprove existence for the full allowed class. The argument also correctly removes tie-breaking as an escape: all relevant allocations are forced.
No repair or additional clarification is needed. The conditional computational question has no mechanism to which it can apply. The proof establishes impossibility, rather than a polynomial-time or hardness classification of a separate optimization problem.
⟨structured verdict parsed — rendered above⟩
Tier: machine-proved, adversarially reviewed by an independent model. No human has checked this proof. Generated from pipeline artifacts by tools/atlas_build.py.