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Analyze fair division of four slots between two artists

Written answer2 deliveries7 min 11 s in total$0.23 in total

Step 1 of 2

The request

Analyze fair division of four slots between two artists

Read the full request

Study a fictional fair division problem, directly as text; no code, files, tools or external research. Four indivisible exhibit slots a,b,c,d must be allocated to two artists A and B, exactly two slots each. Utilities are additive, with no payments. A values (a,b,c,d)=(8,5,4,1); B values them (2,6,3,9). Enumerate all six feasible allocations, giving A's bundle, B's bundle and both utilities. An allocation Pareto-dominates another if both utilities are at least as high and one is strictly higher. Determine the complete Pareto frontier, every allocation maximizing total utility, and every allocation maximizing the product of utilities. An allocation is envy-free if each artist values their own bundle at least as highly as the other's using their OWN valuations; classify all six. A curator claims: "Giving a,b to A and c,d to B is the unique Pareto-optimal allocation and maximizes both total utility and product." Audit every part of that claim. State the limits of the conclusion: these numbers are hypothetical preferences, not evidence about real people.

The journey

  1. Read the requestTurned it into a list of things it would have to prove before calling the work done.
  2. Did the workPlanned the pieces, built them and checked the result as it went.
  3. DeliveredThe answer is written below.

The result

{ "table": [ { "allocation": 1, "A_bundle": "{a,b}", "B_bundle": "{c,d}", "utilities": [ 13, 12 ], "total": 25, "product": 156, "envy_free": true }, { "allocation": 2, "A_bundle": "{a,c}", "B_bundle": "{b,d}", "utilities": [ 12, 15 ], "total": 27, "product": 180, "envy_free": true }, { "allocation": 3, "A_bundle": "{a,d}", "B_bundle": "{b,c}", "utilities": [ 9, 9 ], "total": 18, "product": 81, "envy_free": false }, { "allocation": 4, "A_bundle": "{b,c}", "B_bundle": "{a,d}", "utilities": [ 9, 11 ], "total": 20, "product": 99, "envy_free": true }, { "allocation": 5, "A_bundle": "{b,d}", "B_bundle": "{a,c}", "utilities": [ 6, 5 ], "total": 11, "product": 30, "envy_free": false }, { "allocation": 6, "A_bundle": "{c,d}", "B_bundle": "{a,b}", "utilities": [ 5, 8 ], "total": 13, "product": 40, "envy_free": false } ], "dominance_relations": [ "1→3", "1→4", "1→5", "1→6", "2→3", "2→4", "2→5", "2→6", "3→5", "3→6", "4→3", "4→5", "4→6" ], "pareto_frontier": [ 1, 2 ], "max_total": { "allocations": [ 2 ], "value": 27 }, "max_product": { "allocations": [ 2 ], "value": 180 }, "curator_audit": { "allocation_1_is_pareto_optimal": { "status": "true", "reason": "Allocation 1 is nondominated." }, "allocation_1_is_uniquely_pareto_optimal": { "status": "false", "reason": "Allocation 2 is also nondominated." }, "allocation_1_maximizes_total": { "status": "false", "reason": "25 < 27 for allocation 2." }, "allocation_1_maximizes_product": { "status": "false", "reason": "156 < 180 for allocation 2." }, "overall_claim": { "status": "false" } }, "limitation": "All conclusions use only the supplied hypothetical preferences and are not evidence about real people." }
Time3 min 41 s
Cost$0.15
Finished2026-10-03

Step 2 of 2

The request

B's value for d drops from 9 to 0, affecting exhibit-slot division results

Read the full request

Update only the numerical analysis of the earlier fictional exhibit-slot division after ONE change: B now values d at 0 instead of 9. No other preference or rule changes. Four indivisible slots a,b,c,d, exactly two per artist, additive utilities, no payments; A=(8,5,4,1), B=(2,6,3,0). Return the updated six-allocation table (both bundles, utilities, total, product, envy-free), complete Pareto frontier, ALL total-utility maximizers and ALL product maximizers. State precisely which conclusions changed from the earlier result: frontier A=ab or ac; unique total/product maximizer A=ac; envy-free A=ab,ac,bc. Use each artist's own valuations to test envy. Do not repeat the background or curator narrative, invent preferences, or introduce payments. Direct text only; no tools, files, code or research.

The journey

  1. Read the requestTurned it into a list of things it would have to prove before calling the work done.
  2. Did the workPlanned the pieces, built them and checked the result as it went.
  3. DeliveredThe answer is written below.

The result

| A bundle | B bundle | Utilities (own; other) | Total | Product | Envy-free? | |---|---|---|---:|---:|---| | ab | cd | A: 13 vs 5; B: 3 vs 8 | 16 | 39 | No: B envies ab | | ac | bd | A: 12 vs 6; B: 6 vs 5 | 18 | 72 | Yes | | ad | bc | A: 9 vs 9; B: 9 vs 2 | 18 | 81 | Yes | | bc | ad | A: 9 vs 9; B: 2 vs 9 | 11 | 18 | No: B envies bc | | bd | ac | A: 6 vs 12; B: 5 vs 6 | 11 | 30 | No: both envy the other bundle | | cd | ab | A: 5 vs 13; B: 8 vs 3 | 13 | 40 | No: A envies ab | Pareto frontier: A=ab, ac, or ad, with utility vectors (13,3), (12,6), and (9,9). The other allocations are dominated: bc by ab/ac/ad, bd by ac/ad, and cd by ad. All total-utility maximizers: A=ac and A=ad, tied at 18. All product maximizers: A=ad uniquely, with product 81. Envy-free allocations: exactly A=ac and A=ad. Changes from the earlier conclusions: the frontier expands from ab or ac to ab, ac, or ad; the unique total maximizer ac becomes a tie between ac and ad; the unique product maximizer changes from ac to ad; and envy-free allocations change from ab, ac, bc to ac, ad.
Time3 min 30 s
Cost$0.07
Finished2026-10-03

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