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Econ-591 · economics · Economics

Context dependence of social preferences

LLM status Open unverified

Statement

Original identifier: B4 (atlas item 64). Classification: Social preference identification research program. Original tractability label: Medium (editorial, not a complexity result).

Definitions and assumptions

Let $G$ specify an extensive-form game, monetary payoff vector $\pi$, information, repetition, anonymity, and framing. For $n\geq2$ players, a benchmark inequity-aversion representation is \[ U_i(\pi)=\pi_i-\frac{a_i}{n-1}\sum_{j\ne i}(\pi_j-\pi_i)_+ -\frac{b_i}{n-1}\sum_{j\ne i}(\pi_i-\pi_j)_+, \] where $(z)_+=\max\{z,0\}$, $a_i\geq b_i\geq0$, and $b_i<1$. Reciprocity requires additional, explicitly specified beliefs about intentions; altruism permits utility from others' payoffs without inequity penalties. Let $\theta_i$ collect the declared social-preference parameters. A response model maps utilities and beliefs to observed actions, incorporating prespecified decision noise.

Formal research question

For a declared collection $\mathcal G$ of laboratory and real-world interactions, determine whether one context-invariant latent preference distribution $F_\theta$ and correctly specified context-dependent beliefs can reproduce all action distributions: \[ \forall G\in\mathcal G:\quad P_G(A\in E)=\int P_G(A\in E\mid\theta)\,dF_\theta(\theta) \] for every measurable action event $E$. Identify $F_\theta$ or characterize its observationally equivalent set. Compare the common-parameter model with a prespecified alternative $F_{\theta\mid G}$, using out-of-sample predictive loss and equivalence tolerances rather than unrestricted contextual fitting.

Interpretation and scope

An action can change with context even when preferences remain fixed: beliefs about reputation, future interaction, or another person's intentions may change. Conversely, stable dictator allocations need not establish stable reciprocity. Games must therefore vary payoff inequality, efficiency, intention, and strategic incentives separately wherever feasible. Repeated within-person designs require controls for learning and carryover, and cross-cultural comparisons need common measurement and comprehensible instructions. The inequity-aversion formula is only a benchmark submodel; it is not a mathematical definition covering fairness, altruism, and reciprocity simultaneously. A successful empirical program would identify which contextual changes shift beliefs and which require preference variation, with uncertainty from both stages. Where models cannot be distinguished, reporting an identified set is a substantive conclusion. Recommendations for organizational design additionally require a specified welfare aggregator; willingness to punish is not itself proof that punishment improves welfare.

Source audit and status

The three author-year entries resolve to inequity aversion [S1], experimental social-preference discrimination [S2], and altruistic punishment [S3]. Numbered link [38] is a different Fehr--Gächter paper, the 2000 reciprocity review [S4], and is corrected explicitly. These sources provide models and evidence, not a universal stable social-preference theorem. Context-invariant identification across the chosen game collection remains an editorial research objective whose status must be established for each application.

References

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  1. Ernst Fehr and Klaus M. Schmidt (1999). A Theory of Fairness, Competition, and Cooperation. Quarterly Journal of Economics 114(3), 817--868. https://doi.org/10.1162/003355399556151

    Source role: Inequity-aversion model. Locator: Abstract and article metadata.

  2. Gary Charness and Matthew Rabin (2002). Understanding Social Preferences with Simple Tests. Quarterly Journal of Economics 117(3), 817--869. https://doi.org/10.1162/003355302760193904

    Source role: Experimental discrimination among social-preference models. Locator: Abstract and article metadata.

  3. Ernst Fehr and Simon Gächter (2002). Altruistic Punishment in Humans. Nature 415, 137--140. https://doi.org/10.1038/415137a

    Source role: Evidence on punishment and cooperation. Locator: Abstract and article metadata.

  4. Ernst Fehr and Simon Gächter (2000). Fairness and Retaliation: The Economics of Reciprocity. Journal of Economic Perspectives 14(3), 159--181. https://www.aeaweb.org/articles?id=10.1257/jep.14.3.159

    Source role: Original numbered link [38]; reciprocity review. Locator: Abstract and article metadata.

Common conventions: Source mathematical conventions

Unless an entry states otherwise, agent and firm sets are finite. State and action spaces are measurable, strategies and outcome maps are measurable, probability laws are specified, and expectations exist for the targets under discussion. Infinite-horizon discount factors lie in $(0,1)$ and discounted objectives are finite. Norms, tolerances and complexity input sizes must be specified for computational comparisons. Constants or primitives that appear locally are inputs, not empirical facts asserted by this catalogue.

Identification. A statistical model $m\in\mathcal M$ implies an observable law $P_m$ and target $\theta(m)$. At law $P$, its identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Point identification holds when this set is a singleton, or equivalently when $P_{m_1}=P_{m_2}$ implies $\theta(m_1)=\theta(m_2)$ for all compatible models. Sharp bounds mean bounds attained, or approached when the boundary is not attained, by compatible models. Writing this set is a definition, not a solution: the substantive task is to establish informative restrictions and derive or estimate the set. An unrestricted-confounding model may give only trivial bounds.

Inference. Identification, estimability and valid uncertainty quantification are separate questions. Fix $\alpha\in(0,1)$. A confidence procedure $C_n$ over a declared law class $\mathcal P$ has asymptotic uniform coverage at level $1-\alpha$ when \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P}\Pr_P\{\theta(P)\in C_n\}\geq1-\alpha. \] For set-identified targets the coverage event must be defined separately, for example coverage of the full identified set. If an entry uses identification-indexed models instead of uniquely indexed laws, the same coverage convention is applied over those models. Pointwise limits do not establish uniform validity, and asymptotic validity does not guarantee finite-sample precision. Sample sizes, dependence structures and weak-identification sequences are part of each application.

Counterfactuals. $Y_i(a)$ is a potential outcome under a specified intervention, including its timing, implementation and versions. It is not $Y_i$ conditional on voluntarily choosing $a$. A full intervention vector permits interference; shorthand scalar treatment notation does not silently impose no interference. Assignment, selection, attrition, anticipation and measurement must be specified. A claim about an instrument's local effect is not automatically a population policy effect.

Economic equilibrium. A counterfactual model includes best responses, constraints, feasibility, market clearing, state transitions and expectations consistent with the stated information. When several equilibria exist, either a declared selector is an input or the target is set-valued. Comparisons across policy regimes must use the stated selection convention. Reduced-form relationships are not presumed invariant after an equilibrium-changing intervention.

Optimization and welfare. Optimization is over the stated feasible set. If attainment is not established, $\sup$ or $\inf$ and $\varepsilon$-optimal choices replace an asserted maximizer or minimizer. For example, with value $V=\sup_{a\in\mathcal A}W(a)<\infty$, an $\varepsilon$-optimal set is $\{a\in\mathcal A:W(a)\geq V-\varepsilon\}$ for $\varepsilon>0$. Benefits, costs, risk and health or environmental outcomes can be aggregated only using declared units and conversion weights. Interpersonal utility weights, the treatment of fiscal transfers and foreign profits, discounting, and the behavioral welfare criterion are explicit inputs. A comparative-static derivative additionally requires existence and appropriate differentiability of the selected counterfactual.

Sources. Local labels [S1], [S2], and so forth restart in every question. Locators identify the relevant abstract, model, section or result at the level actually checked; a bibliographic or abstract-level verification is not represented as inspection of an entire proof. Working papers are distinguished from later publications and changing versions. Source summaries are paraphrased. All URL checks and editorial formulations refer to the audit date stated above.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-591-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-591

Context dependence of social preferences: Joint inequity preferences and the missing copula

The context-invariance question

The target asks whether one latent preference distribution explains behavior across a declared family of contexts, with beliefs and response noise separately specified. In the two-person inequity-aversion submodel, \[ U(\pi_1,\pi_2)=\pi_1-a(\pi_2-\pi_1)_+ -b(\pi_1-\pi_2)_+, \qquad a\ge b\ge0,\quad b<1. \tag{1} \] This is the benchmark introduced by Fehr and Schmidt [fs]; it is not a definition of all altruism or reciprocity. We give a diagnostic identification result in nonstrategic allocation tasks, then show exactly why separate measurements of the two marginal distributions can fail in a new task. All allocations below are deterministic, consequential, anonymous, and use the same money scale. There is no reputation motive, learning, carryover, or context-specific utility term in this submodel.

Two threshold tasks

For $r>0$, offer the choice between $(0,0)$ and $(1,1+1/r)$. Call selection of the second option $A_r=1$. For $0\le s<1$, offer $(1,1)$ versus $(1/(1-s),0)$, and call selection of the second option $B_s=1$. Assume deterministic utility maximization and resolve ties in favor of the named second option.
Lemma: The recorded choices satisfy \[ A_r=\mathbf1\{a\le r\},\qquad B_s=\mathbf1\{b\le s\}. \tag{2} \]
Proof: In the first task the second option has utility $1-a/r$, whereas the first has utility zero. In the second task the unequal option has utility $(1-b)/(1-s)$, whereas the equal option has utility one. The two weak preference inequalities are exactly (2). The declared tie rule includes the threshold itself. ∎
Theorem: If both tasks are administered to the same independently sampled people without altering their preferences, then for every tested pair $(r,s)$, \[ \Pr(A_r=1,B_s=1)=F_{a,b}(r,s). \tag{3} \] Knowing these probabilities on dense threshold sets in $(0,\infty)\times[0,1)$ identifies the entire distribution of $(a,b)$. With only a finite rectangular threshold grid, the joint masses of its half-open cells are identified by successive differences; any within-cell distribution supported on $a\ge b\ge0$, $b<1$, is observationally equivalent for the full tested threshold battery.
Proof: Equation (3) follows directly by multiplying the two indicators in (2). For an arbitrary finite threshold pair, approximate both coordinates from above by tested values. Continuity of probability under decreasing events recovers its CDF value. The same argument as $r\downarrow0$ recovers possible mass at $a=0$, and increasing limits toward $b=1$ or $a=\infty$ recover the endpoint totals. A joint CDF determines its probability measure, proving identification. On a finite grid, inclusion and exclusion of adjacent lower rectangles yields each cell mass. Every threshold indicator is constant on each such cell, including its prespecified endpoint convention, so changing the distribution within a cell cannot change any joint law of the tested indicators. Conversely, those joint laws give exactly the cell masses. Cells incompatible with the parameter inequalities must have zero mass. ∎
This theorem uses a joint within-person battery. Randomly assigning different people to the two task families identifies their marginal distributions, but generally not the joint distribution needed for new contexts. An actual experiment has finite thresholds and finite data; the dense-battery statement is a population identification limit, not an assertion that an infinite experiment is feasible.

A concrete failure of marginal transport

Example: Consider two context-invariant populations. Population $P$ assigns probability $1/2$ to each of $(a,b)=(1,0),(2,1/2)$. Population $Q$ assigns probability $1/2$ to each of $(1,1/2),(2,0)$. Every type satisfies the restrictions in (1). Both populations have the same marginal law of $a$ and the same marginal law of $b$, so all separately sampled threshold tasks have identical choice probabilities.

Now offer $(13/5,18/5)$ versus $(1,0)$, choosing the first at ties. Their utility difference is $8/5-a+b$, so the first is chosen exactly when $a-b\le8/5$. Under $P$, the two differences $a-b$ are $1$ and $3/2$, so everyone chooses it. Under $Q$, they are $1/2$ and $2$, so only half do. There are no utility ties. The new-context prediction therefore differs although both original marginal batteries fit exactly.

This is a copula ambiguity within a stable-preference model. It cannot be interpreted as evidence that preferences changed with context. Likewise, a noisy response rule cannot be ignored: with logit mistakes, one report is not the threshold indicator in (2), and identifying the joint distribution requires the declared noise model or additional repeats.

An exact compatibility test for a general finite battery

Let $\Theta$ be a compact parameter set, and let $k:\Theta\to\mathbb R^m$ be a continuous vector of the battery's fully specified event probabilities. These may include known response noise and known context-dependent beliefs. Let $p$ be their observed population vector. Every context shares a single probability law $F$.
Proposition: A common distribution explains $p$ if and only if \[ p\in\operatorname{conv}\{k(\theta):\theta\in\Theta\}. \tag{4} \] Whenever it exists, one explaining distribution has at most $m+1$ support points. If (4) fails, there is a vector $c$ with $c^{\mathsf T}p>\sup_{\theta\in\Theta}c^{\mathsf T}k(\theta)$, which is a population falsification certificate.
Proof: Every finite convex combination of the kernel vectors is generated by a finite-support distribution. Any convex combination with more than $m+1$ points has an affine dependence: coefficients, not all zero, sum to zero and give a zero weighted vector. Moving the mixing weights along this direction until one first reaches zero preserves the combination and reduces its support. Iteration leaves at most $m+1$ points. Since $k(\Theta)$ is compact, its convex hull is compact too: it is the continuous image of the compact product of $m+1$ copies of $\Theta$ and their weight simplex. General probability integrals are limits of finite convex combinations and hence belong to this compact hull. This proves equivalence and the support assertion. Strict separation of a point from a closed convex set gives the final certificate. ∎

Remaining gap

The threshold theorem identifies inequity parameters under strong but transparent allocation-task assumptions; the compatibility test handles any supplied continuous response kernel. Neither establishes stable reciprocity beliefs, distinguishes all social-preference theories, or separates preference variation from misspecified noise. A rejected common kernel can reflect any of these failures. New-context prediction should therefore use the joint identified set and a fixed evaluation loss, while claims about punishment or organizational welfare require a separate welfare criterion.

Completion Estimate

57%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt identifies joint inequity-aversion parameters through within-person threshold tasks and gives finite-battery compatibility and falsification certificates, with a concrete failure of marginal transport. Full context-invariant social-preference assessment still requires credible beliefs and noise models, discrimination of reciprocity and altruism, and held-out prediction against prespecified context-dependent alternatives.

References

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