--
Equilibrium
--
Lying-averse
--
Deception-averse
--
Inference error
--
Avg. welfare

Game Log

click to expand

Fig. 1 — Utility Parameter Distributions

Note. Marginal distributions of the four augmented-utility parameters (Choi, Lee & Lim 2025, §5).
cl : lying cost — penalises literal lies (m ≠ θ, Sobel Def. 3).
cd : deception cost — penalises belief distortion (Sobel Def. 4).
α : CRRA risk-aversion coefficient — composition-based from risk-type proportions.
β : other-regarding (altruism) weight — Normal(0.1, 0.3).
Source distributions : cl, cd ~ LogNormal(μ, σ=1) where μ is the log-scale location parameter configurable via sidebar sliders; each subplot annotation shows the generating distribution and the observed sample mean (x̄).

Fig. 2 — Joint (cl, cd) Distribution

Note. Joint scatter of cl and cd per agent, coloured by behavioural classification (Fig. 5 taxonomy).
High cl → lying-averse (deviates from GL equilibrium).
High cd → deception-averse (deviates from BT equilibrium).
Equal-axis scaling preserves the geometric relationship between the two cost dimensions.

Fig. 3 — Strategy Distribution — BT

Note. Bad-type truth-telling game (BT).
The bad type (θ = 0) sends m = θ with probability v = P(m=1|θ=0).
Unique equilibrium: v* = 1 (dashed line) — the bad type always tells the truth.
This is deceptive: it shifts the receiver’s type belief (Prop. 1, Choi+ 2025).
Deviations below v* = 1 are driven by cd (Prop. 3).

Fig. 4 — Strategy Distribution — GL

Note. Good-type lying game (GL).
The good type (θ = 1) sends m = 0 with probability w = P(m=0|θ=1).
Unique equilibrium: w* = 0 (dashed line) — the good type lies to reveal its type.
This is non-deceptive: it does not distort the receiver’s type belief (Prop. 2, Choi+ 2025).
Deviations above w* = 0 are driven by cl (Prop. 4).

Fig. 5 — Agent Type Proportions

Note. Behavioural classification per the Fig. 5 taxonomy in Choi+ 2025.
Top section — configured risk-attitude composition (α < 0 risk-loving, α ≈ 0 risk-neutral, α > 0 risk-averse).
Bottom section — observed behavioural classification:
Equilibrium — follows both BT and GL equilibria.
Lying-averse — follows BT, deviates in GL due to cl.
Deception-averse — follows GL, deviates in BT due to cd.
Inference error — deviates in both environments.

Fig. 6 — Equilibrium Regions (cl vs cd)

Note. Equilibrium-region map in (cd, cl) space following Props. 3–4 (Choi+ 2025).
Regions:
Full reputation — above solid boundary.
Partial — between the two boundaries.
No reputation — below dashed boundary.
Boundaries:
Solid line: cl = 0.8 cd + 0.2 (full / partial).
Dashed line: cl = 0.3 cd (partial / none).
Agent classification:
Equilibrium Lying-averse Deception-averse Inference error
Position reveals which cost dimension drives deviation from equilibrium.

Fig. 7 — Reputation Dynamics (λ Trajectory)

Note. Average reputation belief (λ) across N periods, showing how type inference evolves as state is carried between periods.
BT : bad-type truth-telling — λ typically rises as the receiver updates beliefs from truthful signals.
GL : good-type lying — λ trajectory depends on how lies interact with receiver inference.
Shaded band shows ±1 s.d. across agents. Period weights (xt) shown as dashed grey line on secondary axis.

Game System Architecture

Complete architecture of the multi-agent reputation game system. Click "Edit in draw.io" to modify the diagram.

Edit in draw.io
Configuration n agents · risk composition · cost distributions · x₂/x₁ · pb · ε Population Generation cl, cd ∼ LogNormal(μ, σ) via Gaussian copula α ∼ Composition-based (risk types) · β ∼ Normal (altruism) BT Environment Bad-type Truth-telling Strategic = Bad type · Behavioral = truth-teller Eq: tell truth at θ=0 (Prop. 1: deceptive truth) Deviation driven by cd (deception aversion) GL Environment Good-type Lying Strategic = Good type · Behavioral = always m=1 Eq: lie at θ=1 (Prop. 2: non-deceptive lie) Deviation driven by cl (lying aversion) Bayesian Belief Update λ(m, θ, v) = Pr(τ=G | m, θ) Receiver: a(m) = E[θ | m] Bayesian Belief Update λ(m, θ, w) = Pr(τ=G | m, θ) Receiver: a(m) = E[θ | m] Strategy (Prop. 3) v = P(m=1 | θ=0) max EUᵃ = EU − cl·𝟙{lie} − cd·D Strategy (Prop. 4) w = P(m=0 | θ=1) max EUᵃ = EU − cl·𝟙{lie} − cd·D Round Execution (N-Period Game) Period 1..N−1: θₜ → mₜ → [ε] → aₜ → λₜ carried → next Period N: myopic play — no future reputation (xₙ₊₁=0) Weights xₜ = ratio^(t−1)/(N−1), geometric from 1 to ratio Behavioral Classification (Figure 5) Equilibrium (BT✓ GL✓) | Lying-averse (BT✓ GL✗) | Deception-averse (BT✗ GL✓) | Inf. Error Welfare Analysis & Visualization Sobel Def 3: Lie Def 4: Dec Choi+ Sec 5 EUᵃ

Lie (Sobel 2020, Definition 3)

A message m is a lie if and only if m ≠ θ — the sender sends a message that does not match the true state of the world.

m ≠ θ ⇒ lie

Deception (Sobel 2020, Definition 4)

A message m is deceptive if it shifts the receiver's type belief away from truth, compared to the alternative message.

D(m,θ) = |λ(m,θ) − λ(mⁿ,θ)| > 0

Augmented Utility (Choi+ 2025, §5)

Agents maximize material payoff minus honesty costs: lying cost (cl) for literal lies and deception cost (cd) proportional to belief distortion.

EUᵃ = EU − cl·𝟙{m≠θ} − cd·D(m,θ)

Deceptive Truth-telling (Prop. 1)

In BT, the bad type tells the truth to build reputation. The message is literally true (m=θ, not a lie) but deceptive (shifts type beliefs).

m = θ (truth) ∧ D > 0 (deceptive)

Non-deceptive Lying (Prop. 2)

In GL, the good type lies to reveal type. The message is a literal lie (m≠θ) but not deceptive (doesn't distort type beliefs).

m ≠ θ (lie) ∧ D = 0 (non-deceptive)

Behavioral Classification (Fig. 5)

Agents are classified by equilibrium adherence: Equilibrium (both), Lying-averse (BT✓ GL✗), Deception-averse (BT✗ GL✓), Inference error (neither).

BT × GL → {Eq, LA, DA, IE}

Glossary & Reference

Abbreviations & Indices

TermFull NameDescription
BTBad-type Truth-tellingEnvironment where the strategic sender is the bad type and the behavioral type tells the truth. Equilibrium strategy: tell truth in period 1 to build reputation (deceptive truth-telling, Proposition 1).
GLGood-type LyingEnvironment where the strategic sender is the good type and the behavioral type always sends m=1. Equilibrium strategy: lie in period 1 to reveal type (non-deceptive lying, Proposition 2).
clLying costCost incurred when an agent sends a message m ≠ θ (literal lie). From Sobel (2020) Definition 3. Drawn from LogNormal(μ, σ).
cdDeception costCost incurred proportional to how much a message shifts the receiver's type belief away from truth. Based on Sobel (2020) Definition 4. Drawn from LogNormal(μ, σ).
αRisk aversionCRRA (Constant Relative Risk Aversion) parameter. α < 0: risk-loving, α ≈ 0: risk-neutral, α > 0: risk-averse.
βAltruism parameterWeight on others' welfare. β > 0: altruistic, β = 0: selfish, β < 0: spiteful. Drawn from Normal distribution.
θState of the worldBinary state θ ∈ {0, 1} drawn uniformly in each period. The sender privately observes θ.
mMessageBinary message m ∈ {0, 1} sent by the sender to the receiver.
aActionReceiver's action a ∈ [0, 1], chosen to minimize expected quadratic loss given beliefs.
λType beliefPosterior probability that the sender is the good/behavioral type: λ(m, θ) = Pr(τ=G | m, θ). Updated via Bayes' rule.
vBT mixing parameterv = P(m=1 | θ=0) in BT. Probability the bad type lies when state is 0. Equilibrium: v=0 (truth-telling).
wGL mixing parameterw = P(m=0 | θ=1) in GL. Probability the good type lies when state is 1. Equilibrium: w=1 (lying).
x1, x2Period weightsWeights on period 1 and period 2 payoffs. High x2/x1 ratio creates strong reputation-building incentive.
pbBehavioral priorPrior probability that the sender is the behavioral (non-strategic) type. Default: 0.5.
εMiscommunication rateProbability that a sent message is flipped in transit (0 → 1 or 1 → 0). Models information noise.
$EU^a$Augmented expected utility$EU^a(m|\theta) = EU(m|\theta) - c_l \cdot \mathbb{1}\{m \neq \theta\} - c_d \cdot D(m,\theta)$. From Choi et al. (2025) Section 5.
$D(m,\theta)$Deception measure$|\lambda(m,\theta) - \lambda(m^n,\theta)|$. How much the chosen message shifts type beliefs compared to the alternative. From Sobel (2020) Definition 4.
KDEKernel Density EstimationNon-parametric method to estimate probability density functions. Used to smooth histograms in the distribution plots.
CRRAConstant Relative Risk AversionUtility function $u(x) = \frac{x^{1-\alpha}}{1-\alpha}$. Widely used in behavioral economics.
Prop. 1Deceptive truth-tellingIn BT, when x2/x1 is large, the bad type tells truth to build reputation. The truth is (a) literally true, (b) deceptive w.r.t. type.
Prop. 2Non-deceptive lyingIn GL, when x2/x1 is large, the good type lies to reveal type. The lie is (a) literally false, (b) not deceptive w.r.t. type.
Prop. 3BT equilibrium regionsCharacterizes (cl, cd)-space: full/partial/no reputation building in BT. Deviation driven by deception aversion (cd).
Prop. 4GL equilibrium regionsCharacterizes (cl, cd)-space: full/partial/no reputation building in GL. Deviation driven by lying aversion (cl).
Fig. 5Classification logicCross-tabulation of BT and GL behavior to identify agent type: equilibrium play, lying-averse, deception-averse, or inference error.

Mathematical Notation

ExpressionMeaning
$U_P = -\sum_i x_i(a_i - \theta_i)^2$Public/receiver payoff. Quadratic loss from action deviating from true state.
$U_G = -\sum_i x_i(a_i - \theta_i)^2$Good type payoff. Aligned with receiver — wants accurate actions.
$U_B = -\sum_i x_i(a_i - 1)^2$Bad type payoff. Always prefers actions close to 1 regardless of state.
$\lambda(m,\theta) = \Pr(\tau{=}G \mid m,\theta)$Posterior type belief. Updated from prior $p_b$ using Bayes' rule given message and state.
$D(m,\theta) = |\lambda(m) - \lambda(m^n)|$Sobel deception measure. Difference in type beliefs between chosen and alternative message.
$EU^a = EU - c_l \cdot \mathbb{1}\{m \neq \theta\} - c_d \cdot D$Augmented utility. Material payoff minus lying cost (if lie) minus deception cost (proportional to $D$).

Plot Descriptions

1

Utility Parameter Distributions

Four sub-plots showing the distribution of each agent parameter: cl (lying cost), cd (deception cost), α (risk aversion), β (altruism). Histograms with KDE curves. Shows the population heterogeneity driving different behavioral responses to the game environments.

2

Joint (cl, cd) Distribution

Scatter plot of each agent's lying cost vs deception cost, colored by behavioral classification. Reveals the relationship between cost parameters and equilibrium behavior. Agents with high cd tend to deviate in BT (deception-averse); agents with high cl tend to deviate in GL (lying-averse).

3

Strategy Distribution — BT / GL

Histograms of truth-telling probability across agents. In BT, equilibrium predicts p=1.0 (full truth); in GL, equilibrium predicts p=0.0 (full lying). The dashed line marks the equilibrium prediction. Deviations from the predicted value reveal lying- or deception-averse behavior.

4

Agent Type Proportions

Horizontal bar chart showing two sets of proportions: (1) Risk attitude composition (risk-loving / neutral / averse) as configured, and (2) Behavioral classification (equilibrium / lying-averse / deception-averse / inference error) as inferred from game outcomes using Figure 5 logic.

5

Equilibrium Regions (cl vs cd)

Maps the (cd, cl)-space into three regions from Propositions 3 & 4: full reputation building (agents follow equilibrium), partial reputation building (mixed strategies), and no reputation building (agents deviate). Each agent is plotted as a dot colored by classification. Boundary lines approximate the theoretical thresholds.

Source Papers

PaperKey Contributions Used
Choi, Lee & Lim (2025)
"The Anatomy of Honesty: Lying Aversion vs. Deception Aversion"		Two-period reputation game model (Section 2). BT and GL environments (Section 3). Deceptive truth-telling & non-deceptive lying (Props. 1-2). Augmented utility with dual honesty costs (Section 5). Equilibrium characterization in (cl, cd)-space (Props. 3-4). Classification logic (Figure 5).
Sobel (2020)
"Lying and Deception in Games"
Journal of Political Economy		Formal definitions of lying (Def. 3: m≠θ) and deception (Def. 4: belief manipulation measure). Theoretical framework distinguishing literal falsehood from strategic belief distortion. Foundational distinction between lying aversion and deception aversion.