| paper | d4-journals-geb-2013-0051 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | medium |
| authors would recognise it | yes |
other→P discrete result → its continuous mirror
Answer: Polynomial-time solvable. · discrete side: other · adjudication: proved
full proof & adversarial review →
Proposition 1
statement extracted from the paper’s text layer
Given rational \(p,r,\beta\) and a finite-support risk-aversion distribution \((a_j,\mu_j)_{j=1}^{\tau}\), where \(a_j>0\), \(\mu_j\ge 0\), and \(\sum_j\mu_j=1\), compute the unique type-level equilibrium investments under PRO and their average \(̅s=\sum_j\mu_js_j\), exactly or to additive error \(\epsilon\). Each type chooses \(s_j\ge 0\) to maximize the paper's CARA expected utility, with success return \(rs_j\) and bankruptcy award \(\beta s_j\).
A unit-mass population of investors has finitely many risk-aversion types \(a_j\), with mass \(\mu_j\) of each type. Each type chooses investment \(s_j\ge 0\); under PRO, its bankruptcy award is \(\beta s_j\). The task is to compute the unique equilibrium vector and average investment \(\sum_j\mu_js_j\).
The type masses leave each type's equilibrium action unchanged and enter only in the average investment, so the mirror adds little population-dependent computation.
fatal: False
The mirror covers Proposition 1's PRO equilibrium characterization. It leaves Propositions 2 and 3 on the mixture rules, Theorems 1 through 4 on investment and welfare comparisons, and the remaining welfare analyses aside.
The strongest mirror I believe in is narrow: computing the investment equilibrium under PRO for a society with many investors but relatively few risk-aversion types. The paper has no named NP-hardness, P, or parameterized-complexity result. My anchor is instead a named, explicit equilibrium characterization.
Lead anchor: Proposition 1, proved in this paper. For the PRO investment game, if \(\ln\!\left(\frac{pr}{(1-p)(1-\beta)}\right)\le 0\), the unique dominant-strategy equilibrium is \((0,\ldots,0)\). Otherwise, the unique dominant-strategy equilibrium has \(s_i^*=\frac{1}{a_i(r+1-\beta)}\ln\!\left(\frac{pr}{(1-p)(1-\beta)}\right)\) for each investor \(i\). There are no other Nash equilibria, and the strategies are strictly dominant.
Here is the continuous problem I would hand to a prover: Continuous-Type PRO Equilibrium Computation. There is a unit mass of investors with a finite support of risk-aversion types \((a_j,\mu_j)_{j=1}^{\tau}\), where each \(a_j>0\), each \(\mu_j\ge 0\), and \(\sum_j\mu_j=1\). The common parameters \(p,r,\beta\) and all \(a_j,\mu_j\) are rational inputs. A type-\(j\) investor chooses an investment \(s_j\ge 0\); success yields net return \(rs_j\), while under PRO the bankruptcy award is \(\beta s_j\). Each investor maximizes the paper’s CARA expected utility with risk aversion \(a_j\). The task is to output the unique equilibrium investment for every supported type and the average per-investor investment \(\bar s=\sum_j\mu_js_j\). A solution is the equilibrium vector, given exactly or approximated to additive error \(\varepsilon\).
Writing \(H=\sum_j\mu_j/a_j\), the solution is \(s_j=0\) for all types when \(pr\le(1-p)(1-\beta)\). Otherwise, \(s_j=\frac{1}{a_j(1+r-\beta)}\ln\!\left(\frac{pr}{(1-p)(1-\beta)}\right)\), and \(\bar s=\frac{H}{1+r-\beta}\ln\!\left(\frac{pr}{(1-p)(1-\beta)}\right)\). This should be tractable: the equilibrium is a closed form involving a finite sum and a logarithm, with the zero-investment threshold test reducible to rational arithmetic.
The regime is many small funds investing in a pooled, capital-intensive venture, with standardized risk mandates creating a modest number \(\tau\) of shared types among a much larger number \(N\) of funds. Each type has the same risk aversion and all other model parameters are common; its mass is its fraction of the investor population. Aggregate investment scales with \(N\), while \(\bar s\) is per investor. This is a direct high-multiplicity regime for the paper: its authors explicitly allow investors to represent investment funds, and risk aversion is the model’s only agent-specific attribute. Under PRO, individual incentives do not depend on the other investors’ choices, so the equilibrium formula applies to every finite population composition as well as to its continuous population description.
The mirror covers Proposition 1’s PRO equilibrium only. It does not cover the paper’s comparisons among bankruptcy rules or its welfare results. Further questions include how quickly finite populations approximate other rule families’ continuum equilibria, and whether constrained rules retain unique computable equilibria across risk-aversion distributions.
The weakest point is that this is an equilibrium characterization, not a complexity-class result; the paper does not establish a hard computational problem, and the PRO case makes strategic interaction especially simple. So this is a credible population-continuum mirror, but a modest contribution to a programme about the complexity landscape.
Proposition 1 is the best anchor, but the proposed mirror adds almost no population-dependent computation. The paper already gives the equilibrium for an arbitrary finite list of risk-aversion parameters. Let
\[
K=\frac{\max\!\left\{0,\ln\!\left(\frac{pr}{(1-p)(1-\beta)}\right)\right\}}{1+r-\beta}.
\]
Then investor \(i\) chooses \(s_i=K/a_i\). For a continuous society with finitely many risk-aversion types, the type-level choices are the same, \(s_j=K/a_j\), and the average investment is just \(\bar{s}=K\sum_j\mu_j/a_j\). The threshold is a rational comparison; the only non-rational output is a logarithm that can be approximated. Grouping identical investors by type compresses the input, but creates no new equilibrium problem or algorithmic bottleneck.
The many-fund scenario is credible, and the paper itself invites it by allowing each agent to represent an investment fund. The objection is not that high multiplicity is implausible. It is that under PRO, each investor’s payoff is independent of everyone else’s investment. The population distribution affects neither the best response nor the equilibrium threshold; it enters only when averaging the already-known actions. Even a continuous distribution of risk aversion would merely push that distribution through \(s(a)=K/a\), with any computation depending on how the distribution is encoded.
One could instead study how an atomless risk-aversion distribution changes comparisons among PRO, EA, and EL. That may be an interesting extension, but it is no longer the narrow equilibrium-computation mirror offered here. The paper already characterizes the finite games across heterogeneous risk-aversion profiles, while a version with new aggregate-dependent returns or other population interactions would change the game that Proposition 1 describes. Such work looks more like equilibrium analysis than the complexity landscape ChoCo is trying to chart.
So the honest negative case is limited: this is a valid continuous-population mirror, but a poor computational contribution. I cannot make the stronger universal claim that no worthwhile mirror exists in any scenario—the fund-pool regime is sensible, and a programme might value a tractable benchmark. The case against rests on the fact that this anchor’s continuous version appears to add type-weighted reporting, not a distinct population-computational question.
The adversarial triple: the proponent anchors on up to three named results; the opponent sees that case and must defeat every anchor; the judge decides which case convinced it. These are the pipeline’s own outputs, generated by tools/triple_run.py — no human edited them. The paper’s own text is not reproduced here beyond the quoted statement above.