2.1. Individual’s preferences

The present model can be viewed as an extension of the lying-aversion framework of Gneezy et al. (Reference Gneezy, Kajackaite and Sobel2018) by explicitly incorporating prosocial motivations into the structure of intrinsic lying costs. Specifically, whereas Gneezy et al. (Reference Gneezy, Kajackaite and Sobel2018) model intrinsic lying costs as functions of individual-specific fixed and outcome-contingent components, the present framework allows these costs to be partially offset when lying generates positive externalities. The analysis focuses on environments in which individuals interact in dyads. Throughout the paper, $P_i$, where $i \in \{1,2\}$, denotes the player occupying role $i$ in the dyad. When the role is irrelevant, I refer simply to a generic individual $i$.

The experiment employs a binary lying game.Footnote ⁴ Each player $i \in \{1,2\}$ privately draws a state $x_i \in \{0,1\}$, with $\Pr(x_i = 1) = 0.2$, and sends a report $r_i \in \{0,1\}$. Their payoffs are interdependent, with both players receiving the monetary payoff:

\begin{equation*} v(r_i, r_j) \;=\; \begin{cases} v_h & \text{if } r_i + r_j \geq 1,\\[2pt] v_l & \text{if } r_i + r_j = 0, \end{cases} \qquad \text{with } v_h \gt v_l \geq 0. \end{equation*}

Because the payoff function is identical for both players and symmetric in the reports, it is written without a player index. The only exception is treatment Sequential-NoExternality, introduced in Section 2.2.3, where $P_2$’s payoff depends only on their own draw, and the payoff function is indexed accordingly. In this context, lies are characterized as Pareto White Lies (Erat & Gneezy, Reference Erat and Gneezy2011) because they benefit both the liar and others.

Individuals’ preferences depend on three elements that determine the willingness to lie or tell the truth. First, they get utility from the monetary payoff $v_i \in \{v_h, v_l\}$ that depends on their report. All else equal, they have extrinsic incentives to report 1 regardless of their actual random draw $x_i$. Second, individuals dislike lying. Lying aversion is represented by some psychological costs ( $c_i$) that they incur when they misreport their random draw, following the standard lying models by Gneezy et al. (Reference Gneezy, Kajackaite and Sobel2018), Abeler et al. (Reference Abeler, Nosenzo and Raymond2019), and Khalmetski and Sliwka (Reference Khalmetski and Sliwka2019).Footnote ⁵ Let these costs, represented by $c_i$, be distributed among the population according to $c_i \sim U[0, \bar{c}]$. Hence, the cumulative density function of $c_i$ is $F(c_i) = \frac{c_i}{\bar{c}}$. The heterogeneity in the psychological lying costs captures the fact that some people are more morally inclined than others and follows the convention introduced by Gneezy et al. (Reference Gneezy, Kajackaite and Sobel2018), Abeler et al. (Reference Abeler, Nosenzo and Raymond2019), and Khalmetski and Sliwka (Reference Khalmetski and Sliwka2019).

Third, I incorporate prosociality into the utility function, drawing on insights from Wiltermuth (Reference Wiltermuth2011), Levine and Schweitzer (Reference Levine and Schweitzer2015), and Janezic (Reference Janezic2020). To capture the heterogeneity in prosocial motivations observed in real populations, I assume that individuals derive utility $\theta_i$ when they benefit another agent with their report, that is, when they generate a positive externality. The prosocial parameter $\theta_i$ is distributed among the population according to

\begin{equation*} \theta_i \sim U[0, \bar{\theta}]. \end{equation*}

This heterogeneity reflects agents’ varying concern for others’ welfare, analogous to heterogeneity in agents’ intrinsic lying costs. I adopt the taxonomy introduced by Sobel (Reference Sobel2020), which distinguishes among lying (making a statement one believes to be false), deception (inducing the receiver to form incorrect beliefs), and damage (the harm inflicted on the receiver by the falsehood). Under this taxonomy, the lies studied here involve neither deception regarding payoff-relevant states of the receiver nor damage: a misreport can only increase the partner’s monetary payoff. The central behavioral assumption of the model is that such non-damaging lies entail a lower moral cost: when an agent’s misreport increases the partner’s payoff, the prosocial benefit, $\theta_i$, partially offsets the intrinsic moral cost of lying. This offsetting is an assumption, not an implication of the taxonomy; its role and testable implications are discussed below and in Section 2.2.

The parameter $\theta_i$ is related to, but distinct from, altruistic preferences in the tradition of Andreoni (Reference Andreoni1990) and Charness and Rabin (Reference Charness and Rabin2002). Under pure altruism, an agent’s utility is increasing in the partner’s material payoff regardless of how that payoff comes about. Here, by contrast, $\theta_i$ never enters utility as a stand-alone benefit from the partner’s payoff: as equation (1) below makes precise, it operates only inside the lying-cost term, reducing the moral cost of a misreport when that misreport is what generates the partner’s gain. An agent with $\theta_i \gt 0$ derives no utility from the partner’s payoff when it is produced by the partner’s own report, whereas a pure altruist would. In this sense the model belongs to the family of moral-justification approaches to dishonesty (Shalvi et al., Reference Shalvi, Dana, Handgraaf and De Dreu2011; Shalvi et al., Reference Shalvi, Gino, Barkan and Ayal2015), in which features of the environment attenuate the psychological cost of lying, rather than to the family of outcome-based social preference models.

For simplicity, I abstract from the possibility that individuals are prosocial toward the experimenter. Although this paper focuses exclusively on positive externalities, where prosociality serves only to reduce the net cost of lying, the framework can be readily extended to allow $\theta_i \lt 0$, thereby capturing settings in which a lie harms others and increases the net psychological cost of deception (as in Gneezy and Kajackaite, Reference Gneezy and Kajackaite2020; Kerschbamer et al., Reference Kerschbamer, Neururer and Gruber2019).

I impose $\bar{\theta} \geq 0$, thereby restricting attention to positive externalities. No upper bound on $\bar{\theta}$ is required for any of the results: the support condition $v_h - v_l + \bar{\theta} \lt \bar{c}$, introduced below, alone guarantees that all lying-cost thresholds derived in Propositions 1–5 are interior, and the treatment rankings in those propositions and in Corollaries 1–2 require only $\bar{\theta} \gt 0$ together with that condition. The parameter values affect the magnitudes of the predicted differences, but not their signs.

A remaining modeling choice concerns when the offset applies: does an agent experience moral relief from any lie that could, in principle, benefit the partner, or only from a lie that actually delivers that benefit? Models of warm-glow giving (Andreoni, Reference Andreoni1990) attach utility to the act of giving itself, independently of its consequences, whereas models of pure altruism attach utility to the partner’s realized outcome regardless of who produced it. I adopt a third position, tailored to the moral-justification interpretation of the offset: a misreport is morally discounted only insofar as it is the action that secures the partner’s gain. A lie told when the partner’s high payoff is already secured, or is being secured by the partner at the same moment, provides no moral justification and therefore no moral relief.Footnote ⁶

To formalize this idea while accommodating treatment-specific information structures, I introduce a justification indicator $I_i \in [0,1]$: the probability, given the agent’s information at the moment of reporting, that their misreport guarantees the partner the high payoff without the partner’s own report having already secured it or securing it simultaneously. A first mover’s lie in a sequential protocol carries full weight, $I_i = 1$, because it settles the joint outcome before the partner acts; a lie submitted simultaneously with the partner’s report carries weight $1 - b$, where $b$ is the probability that the partner reports 1, because with probability $b$ the partner secures the payoff independently; and a lie that cannot affect the partner’s payoff carries no weight. The value of $I_i$ for each treatment is specified in the corresponding subsection. It is worth emphasizing that the model does not assume outcome-based social preferences of the form $U_i = \text{payoff}_i + \alpha \cdot \text{payoff}_j$; the externality enters only by offsetting the intrinsic cost of lying. With these three elements in place, individuals’ preferences are represented by the following utility function:

(1)\begin{equation} U_i(x_i, r_i, r_j) = v(r_i, r_j) - \mathbf{1}_{x_i \neq r_i}(c_i - \theta_i \cdot I_i) \end{equation}

where $v(r_i, r_j) = v_h$ if $r_i + r_j \geq 1$, and $v_l$ otherwise. $I_i$ captures whether lying has prosocial intent and varies by treatment as specified in the following sections. For algebraic clarity in the subsequent proofs, I normalize the low payoff such that $ v_{l} = 0.$ The high payoff $v_{h}$ then represents the net monetary gain from a successful outcome. This normalization is standard and does not affect the model’s qualitative predictions, as all results can be generalized by substituting $\,v_{h} - v_{l}\,$ for $v_{h}$ where necessary.

Before introducing the treatments, it is useful to be explicit about what the offset $\theta_i I_i$ does and does not capture. A reduction in the net cost of lying when the lie benefits the partner is consistent with two distinct motives: a genuine other-regarding concern for the partner’s payoff, or a self-serving use of the positive externality as a justification for a lie the agent is materially tempted to tell anyway (Shalvi et al., Reference Shalvi, Dana, Handgraaf and De Dreu2011; Shalvi et al., Reference Shalvi, Gino, Barkan and Ayal2015). The model deliberately remains agnostic about which motive generates the offset, and the experimental treatments do not distinguish between them: they identify whether, and under what strategic conditions, a positive externality causally increases lying, but not the underlying mechanism. All hypotheses below, and the interpretation of the results, should be read with this scope in mind. Disentangling genuine prosociality from self-serving justification would require additional treatments, for example, varying the magnitude of the externality independently of the liar’s own monetary stake, and is left for future work.

2.2. Treatments

2.2.1. Treatment 1: Sequential

In the main treatment, Sequential, I study a two-stage lying game where players’ lies are substitutes. $P_1$ draws $x_1 \in \mathcal{X}$ and sends a report $r_1 \in \mathcal{X}$ to $P_2$. After observing $r_1$, $P_2$ draws $x_2 \in \mathcal{X}$ and sends a report $r_2 \in \mathcal{X}$. Note that $x_i$ is known only to $P_i$, not to the other player.

In this treatment, the justification indicator is:

(2)\begin{equation} I_i = \begin{cases} r_i & \text{for } P_1 \\ r_i \cdot \mathbf{1}_{r_1 = 0} & \text{for } P_2 \end{cases} \end{equation}

I apply backward induction to analyze the game’s strategic environment. Importantly, in this game, there is no downward lying in equilibrium. An individual who draws $x_i = 1$ and reports $r_i = 0$ incurs the lying cost without receiving either the monetary payoff or the benefit of a positive externality. Thus, in equilibrium individuals only lie if they draw $x_i=0$ by reporting $r_i=1$.

Proposition 1. (P2’s Best Response in Sequential)

After observing $r_1$, $P_2$’s best response is:

• • If $r_1 = 1$: $P_2$ reports $r_2 = 1$ if $x_2 = 1$, and $r_2 = 0$ if $x_2 = 0$

• • If $r_1 = 0$: $P_2$ reports $r_2 = 1$ if $c_2 \lt v_h + \theta_2$

Proof. Recall that I have normalized the low payoff to zero, $v_{l}=0$.

Case 1: $r_{1}=1$. If $x_{2}=0$ and $P_{2}$ reports truthfully ( $r_{2}=0$), their payoff is

\begin{equation*} U_{2} = v(r_{1}=1, r_{2}=0) = v_{h}. \end{equation*}

If instead they lie ( $r_{2}=1$), since $I_{2}=0$ there is no prosocial benefit, so

\begin{equation*} U_{2} = v_{h} \;-\; c_{2}. \end{equation*}

Because $c_{2} \gt 0$, truth-telling ( $r_{2}=0$) strictly dominates lying when $r_{1}=1$.

Case 2: $r_{1}=0$ and $x_{2}=0$. Then

\begin{equation*} U_{2}(r_{2}=0) = v_{l} = 0, \qquad U_{2}(r_{2}=1) = v_{h} - \bigl(c_{2}-\theta_{2}\bigr) = v_{h} - c_{2} + \theta_{2}. \end{equation*}

Hence $P_{2}$ will lie (report $r_{2}=1$) if and only if

\begin{equation*} v_{h} - c_{2} + \theta_{2} \gt 0 \quad\Longleftrightarrow\quad c_{2} \lt v_{h} + \theta_{2}. \end{equation*}

Let $\hat{c}_i(\theta_i)$ be the lying cost threshold where individuals with prosociality $\theta_i$ are indifferent between lying and telling the truth. For $P_2$, when $P_1$ reports 0, I have $\hat{c}_2(\theta_2, r_1 = 0) = v_h + \theta_2$. Hence, the probability that $P_2$ lies after $P_1$ reports 0, conditional on prosociality level $\theta_2$, is:

(3)\begin{equation} \Pr({P_2 lies}|r_1 = 0, \theta_2) = F(\hat{c}_2(\theta_2, r_1 = 0)) = \frac{v_h + \theta_2}{\bar{c}} \end{equation}

To state $P_1$’s problem, let $b_0$ denote $P_1$’s belief that $P_2$ reports 1 after observing $r_1 = 0$. In equilibrium, this belief is determined by Proposition 1: after $r_1 = 0$, $P_2$ reports 1 truthfully when $x_2 = 1$ and lies whenever $c_2 \lt v_h + \theta_2$, so

\begin{equation*} b_0 \;=\; 0.2 + 0.8 \, \frac{v_h + \bar{\theta}/2}{\bar{c}} . \end{equation*}

For convenience, define

\begin{equation*} \lambda \equiv \frac{v_h + \bar{\theta}/2}{\bar{c}} \in (0,1), \end{equation*}

so that $b_0 = 0.2 + 0.8\lambda$.

Proposition 2. (P1’s equilibrium strategy in Sequential)

$P_1$ lies (reports $r_1 = 1$ when $x_1 = 0$) if and only if their lying cost $c_1$ is below the threshold

(4)\begin{equation}\widehat c_1^{\,seq}(\theta_1)\;=\;\theta_1+0.8\,v_h\left(1-\frac{v_h+\overline\theta/2}{\overline c}\right).\end{equation}

Proof. $P_1$ anticipates $P_2$’s equilibrium strategy characterized in Proposition 1. If $P_1$ reports truthfully ( $r_1 = 0$), the joint payoff is $v_h$ whenever $P_2$ reports 1, which occurs with probability $b_0$, so

\begin{equation*} \mathbb{E}[U_1 \mid r_1 = 0] = b_0 v_h. \end{equation*}

If $P_1$ lies, the joint payoff is $v_h$ with certainty, incurring the lying cost $c_1$, and the justification weight is $I_1 = 1$, since the lie determines the joint outcome before $P_2$ acts. Hence,

\begin{equation*} \mathbb{E}[U_1 \mid r_1 = 1] = v_h - c_1 + \theta_1. \end{equation*}

Therefore, $P_1$ lies if and only if

\begin{equation*} v_h - c_1 + \theta_1 \; \gt \; b_0 v_h \quad\Longleftrightarrow\quad c_1 \; \lt \; \theta_1 + (1 - b_0)\, v_h, \end{equation*}

and substituting $1 - b_0 = 0.8(1 - \lambda)$ yields equation 4.

2.2.2. Treatment 2: Sequential-Computer

In a second treatment, Sequential-Computer, I remove $P_1$’s ability to rely on $P_2$’s possibility of lying by imposing $x_2 = r_2$. To do so, in Sequential-Computer, participants in the role of $P_2$ do not have the opportunity to report their draw, but the computer observes the draw and submits the report on their behalf. This procedure is common knowledge among participants. The payoff structure is the same as in Sequential. Even though the computer submits the report, the human participant bears the resulting payoff consequences. Thus, this procedure ensures that $P_1$ faces an objective probability distribution over $r_2$. This feature implies that $b_0 = 0.2$.

In Sequential-Computer, the justification indicator for $P_1$ is simply $I_1 = r_1$ since lying always has the potential to help $P_2$.

Proposition 3. (P1’s equilibrium in Sequential-Computer)

P1 lies if and only if

(5)\begin{equation} c_1 \; \lt \; \theta_1 + 0.8\, v_h . \end{equation}

Proof. P1 faces the objective probability $\Pr(r_2 = 1) = 0.2$. Truth-telling yields $\mathbb{E}[U_1 \mid r_1 = 0] = 0.2\, v_h$. Lying secures $v_h$ with certainty and carries full justification weight, $I_1 = 1$, since P1’s report settles the joint outcome before the automated report is made: $\mathbb{E}[U_1 \mid r_1 = 1] = v_h - c_1 + \theta_1$. P1 lies if and only if $v_h - c_1 + \theta_1 \gt 0.2\, v_h$, that is, $c_1 \lt \theta_1 + 0.8\, v_h$.

Comparing the thresholds in equations (4) and (5) shows that a larger proportion of $P_1$ participants lie when they cannot rely on $P_2$’s willingness to lie. The difference between the thresholds equals $0.8\,v_h\lambda = 0.8\,v_h(v_h + \bar{\theta}/2)/\bar{c}$, is the same for every $\theta_1$, and is positive because $b_0 \gt 0.2$: in Sequential, $P_2$’s propensity to misreport after observing a No increases the expected payoff from truth-telling, thereby weakening $P_1$’s incentive to lie. This result motivates Hypothesis 1.

Hypothesis 1. (Cost avoidance)

In Sequential-Computer, the proportion of $P_1$ participants who lie is higher than in Sequential.

Hypothesis 1 tests whether $P_1$ participants exploit the opportunity to avoid lying costs afforded by Sequential. In Sequential, $P_1$ can leave the lie, and the associated psychological cost, to $P_2$, an option that is removed by design in Sequential-Computer, where $P_2$’s report is truthful by construction. If the data do not support this hypothesis, it would suggest that, within this setup and for the payoff parameters implemented here, intrinsic lying costs are small relative to the value of the externality, so that the incentive to secure the joint payoff dominates the incentive to avoid lying costs. This alternative prediction, characterized by equal lying rates across treatments, can be rationalized within the model under the particular parameter configuration described below.

Corollary 1. (Convergence under large cost heterogeneity)

Let $c_i \sim U[0, \bar{c}]$. As the upper bound $\bar{c}$ becomes large relative to the sum of the monetary upside and the average value of the externality,

\begin{equation*} \frac{\bar{c}}{\,v_h + \bar{\theta}/2\,} \longrightarrow \infty, \end{equation*}

the difference between the two lying-cost thresholds,

\begin{equation*}\widehat c_1^{\,seq-comp}(\theta_1)-\widehat c_1^{\,seq}(\theta_1)\;=\;0.8\,v_h\,\frac{v_h+\overline\theta/2}{\overline c},\end{equation*}

tends to zero. Consequently, the predicted lying rates in the two treatments converge and coincide in the limit. The proof is given in Appendix B.

Corollary 1 highlights that under a uniform distribution $U[0,\bar c]$, increasing $\bar c$ does more than raise the upper bound of potential lying costs—it broadens the support of $c_i$, thereby increasing heterogeneity across individuals. As $\bar{c}$ becomes large, only a vanishingly small fraction of participants find it optimal to exploit the strategic opportunity for cost avoidance, regardless of their $\theta_1$. The threshold gap equals $0.8\,v_h(v_h + \bar{\theta}/2)/\bar{c}$, so the rate of convergence depends on the payoff parameters. Higher values of $v_h$ or $\bar{\theta}$ slow convergence, whereas greater cost heterogeneity $\bar{c}$ accelerates it.

Finally, my framework is based on substitute, rather than complementary, lies, unlike the collaborative lying games surveyed by Leib et al. (Reference Leib, Köbis, Soraperra, Weisel and Shalvi2021). Appendix D analyzes a conjunctive variant of the game in which both players must report 1 to secure $v_h$, as in Weisel and Shalvi (Reference Weisel and Shalvi2015). Requiring both reports changes the first mover’s incentives in two opposing ways. The monetary return to lying becomes $\beta v_h$ instead of $(1 - \beta)v_h$, where $\beta$ is the probability that the partner reports 1 after observing a Yes. Without the first mover’s report, the low payoff is certain, so the free-riding option disappears, which raises the return to lying whenever $\beta \gt 1/2$. At the same time, the justification benefit is diluted from $\theta_1$ to $\beta\theta_1$, because the first mover’s lie helps the partner only when the partner completes it. As Appendix D shows, first movers lie more in the collaborative game than in Sequential only when $\beta \gt 1/2$ and their $\theta_1$ is sufficiently small; for strongly prosocial types, the ranking reverses. The model therefore attributes high dishonesty in collaborative settings to the removal of the free-riding option rather than to a uniform moral sanction against coordination.

2.2.3. Treatment 3: Sequential-NoExternality

In a third treatment, I investigate the role of positive externalities in $P_1$’s decision-making. I use the same structure as in Sequential-Computer, but modify the payoff scheme to eliminate benefits to others. As a result, lies in this treatment are no longer Pareto white lies but instead constitute purely selfish lies. I keep $P_1$’s monetary payoffs identical to those in Sequential-Computer, but make $P_2$’s payoffs depend solely on their own draw: specifically, in Sequential-NoExternality, $P_2$ receives a payoff of $v_h$ if $x_2 = 1$ and $v_l$ otherwise. This variation implies that, in the utility function, $I_1 = 0$, since lying no longer generates any benefit for others. Conceptually, this treatment approximates a setting akin to Gneezy et al. (Reference Gneezy, Kajackaite and Sobel2018), in which the value of prosocial lying is zero.

Proposition 4. (P1’s equilibrium in Sequential-NoExternality)

P1 lies if and only if $c_1 \lt 0.8\, v_h$.

Proof. With $I_1 = 0$, the justification term vanishes. A $P_1$ with $x_1 = 0$ compares $\mathbb{E}[U_1 \mid r_1 = 0] = 0.2\, v_h$ with $\mathbb{E}[U_1 \mid r_1 = 1] = v_h - c_1$, and lies if and only if $v_h - c_1 \gt 0.2\, v_h$, that is, $c_1 \lt 0.8\, v_h$.

Corollary 2. (Sequential-Computer vs. Sequential-NoExternality)

The proportion of P1 participants who lie in Sequential-Computer exceeds that in Sequential-NoExternality.

Proof. From the two thresholds, $\widehat c_1^{\,seq-comp}(\theta_1)=\theta_1+0.8\,v_h$ and $\widehat c_1^{\,seq-noext}=0.8\,v_h$, it follows that $\widehat c_1^{\,seq-comp}(\theta_1)\ge\widehat c_1^{\,seq-noext}$ for any $\theta_1 \gt 0$. Since $\theta_1 \sim U[0, \bar{\theta}]$ with $\bar{\theta} \gt 0$, a positive mass of players satisfies $\theta_1 \gt 0$, so aggregate lying is strictly higher in Sequential-Computer.

Hypothesis 2. (Positive Externality)

In Sequential-NoExternality, the proportion of $P_1$ participants who lie is lower than in Sequential-Computer.

Treatment Sequential-NoExternality isolates prosocial motives. Hypothesis 2 therefore predicts a lower misreporting rate in Sequential-NoExternality, because this treatment removes the prosocial channel. If the data support this hypothesis, it would indicate that participants are more willing to misreport when doing so raises a counterpart’s payoff. Conversely, failure to support the hypothesis would suggest that prosocial considerations exert little influence on misreporting behavior.

While I do not directly compare Sequential and Sequential-NoExternality in my empirical analysis, the theoretical comparison is instructive. The prediction is ambiguous and depends on an individual’s specific prosociality $\theta_i$. The lying threshold in Sequential is higher than in Sequential-NoExternality if and only if

\begin{equation*} \theta_1 \; \gt \; \kappa \;\equiv\; 0.8\,v_h\,\frac{v_h + \bar{\theta}/2}{\bar{c}}. \end{equation*}

For individuals with $\theta_1 \gt \kappa$, the justification channel more than offsets the incentive to rely on $P_2$’s lie. For individuals with $\theta_1 \lt \kappa$, the strategic incentive to free-ride dominates the justification effect, so they lie less than in the purely selfish Sequential-NoExternality context. The cutoff $\kappa$ reappears in the comparison between Sequential and Simultaneous in Section 2.2.4.

Until now I have introduced the main treatments for disentangling lying aversion and prosocial motives. In Sequential-Computer, $P_2$’s report is fixed at 0.2, whereas in Sequential it is subjective, granting $P_1$ greater opportunity to avoid lying costs, while maintaining a prosocial incentive. If lying rates are identical across these two conditions, cost-avoidance effects must be negligible and prosocial motives must fully account for behavior. In Sequential-NoExternality, the prosocial benefit vanishes ( $I_1 = 0$), isolating self-interested lying. Hence, if lying is more frequent in Sequential than in Sequential-NoExternality, positive externalities must be amplifying dishonesty; if it is less frequent, strategic cost avoidance must dominate; and if the rates coincide, the two forces exactly counterbalance. The three sequential treatments therefore differ in exactly two components of the model: the probability that P2 reports 1 after observing $r_1 = 0$, $b_0 = 0.2 + 0.8\,(v_h + \bar{\theta}/2)/\bar{c}$ in Sequential, versus the exogenous value of $0.2$ in Sequential-Computer and Sequential-NoExternality, and the presence of the externality, which is active in Sequential and Sequential-Computer but absent ( $I_1 = 0$) in Sequential-NoExternality. Table 1 summarizes the treatments and indicates which component is isolated by each pairwise comparison.

2.2.4. Treatment 4: Simultaneous

The final treatment, Simultaneous, preserves the payoff structure of the Sequential treatment while eliminating $P_1$’s first-mover advantage. In particular, both players simultaneously choose their report $r_i$, unaware of the other’s decision. As in Sequential, a lie can generate a positive externality only if it changes the joint outcome, but now each player is uncertain whether their misreport will be “effective,” that is, change the joint outcome. That is, it depends on whether the partner would otherwise have reported 0. To capture this, I set

\begin{equation*} I_i \;=\; r_i\;\Pr(r_j=0\mid r_i)\;=\;r_i\,(1-b), \end{equation*}

where $b$ denotes the (symmetric) equilibrium probability that the other player reports 1.

Proposition 5. (Symmetric equilibrium in Simultaneous)

The game has a unique symmetric equilibrium in cutoff strategies. In it, each player $i$ with $x_i = 0$ lies if and only if

\begin{equation*} c_i \; \lt \; (1 - p^{*}) \left( v_h + \theta_i \right), \end{equation*}

where the equilibrium probability that a player reports 1 is

\begin{equation*} p^{*} \;=\; \frac{0.2 + 0.8\lambda}{1 + 0.8\lambda} \;\in\; (0.2,\, 1), \qquad \lambda = \frac{v_h + \bar{\theta}/2}{\bar{c}} , \end{equation*}

and $p^{*}$ is increasing in $\lambda$. The proof is in Appendix B.

Intuitively, each player anticipates that the partner reports 1 with probability $p^{*}$, and this single probability attenuates both incentives to lie by the same factor: the monetary return to a lie is $(1 - p^{*})v_h$, because with probability $p^{*}$ the partner secures the payoff anyway, and the justification weight is $I_i = 1 - p^{*}$, because a lie provides moral cover only when the partner does not independently deliver the benefit.

The comparison with Sequential is immediate. Subtracting the two thresholds gives

\begin{equation*}\widehat c_1^{\,seq}(\theta_1)-\widehat c_1^{\,sim}(\theta_1)\;=\;p^\ast\left(\theta_1-\kappa\right),\qquad\kappa=0.8\,v_h\,\frac{v_h+\overline\theta/2}{\overline c},\end{equation*}

so an individual lies more in Sequential than in Simultaneous if and only if $\theta_1 \gt \kappa$, the same cutoff that governs the comparison between Sequential and Sequential-NoExternality. Averaging over types, aggregate lying is higher in Sequential than in Simultaneous if and only if $\bar{\theta}/2 \gt \kappa$. This is the precise sense in which Hypothesis 3 is directionally ambiguous: the ranking depends on whether the population’s average justification value exceeds $\kappa$.

Hypothesis 3. (Strategic uncertainty)

Conditional on the population’s average justification value, lying is more frequent in Simultaneous than in Sequential if $\bar{\theta}/2 \lt \kappa$, and less frequent if $\bar{\theta}/2 \gt \kappa$.

If the lying rate is higher in Simultaneous, this indicates either (i) that participants misreport primarily out of self-interest, seeking to secure the payoff while avoiding reliance on the partner, or (ii) that the lying-cost offset takes a warm-glow form, attaching to the act of reporting 1 itself rather than to its consequence for the partner, so that strategic uncertainty in Simultaneous does not attenuate the offset.

Table 1 summarizes, for each of the four treatments, the decision nodes, the corresponding payoff structure, and the theoretical predictions. Finally, note that direct treatment comparisons that isolate a single factor are possible only for the following pairs: Sequential vs. Sequential-Computer, Sequential vs. Simultaneous, and Sequential-Computer vs. Sequential-NoExternality. By contrast, in the comparisons Sequential vs. Sequential-NoExternality, Sequential-Computer vs. Simultaneous, and Sequential-NoExternality vs. Simultaneous, multiple factors vary simultaneously, preventing the clean identification of individual causal effects.

Table 1.

Summary of actions, payoffs, and predictions in each treatment

Table 1 Long description

Four treatments are listed with columns for Treatment, Player 1 action, Player 2 action, Payoffs, Prediction, and Isolates comparison. In Sequential, Player 1 reports r1, then Player 2 learns r1 and reports r2; payoffs depend on whether both reports are zero, and the prediction and isolates comparison are both baseline. In Seq-Compa, Player 1 reports r1, then Player 2 learns r1 but the computer makes the report; payoffs follow the same both zero versus otherwise structure, the prediction says Player 1 lies more than in Sequential, and the isolates comparison is cost avoidance versus Sequential by removing reliance on Player 2s lie. In Seq-NoExtb, Player 1 reports r1, then Player 2 learns r1 but the computer makes the report; payoffs are listed separately for Player 1 using the both zero versus otherwise structure and for Player 2 based on whether r2 is zero or one, the prediction says Player 1 lies less than in Seq-Compa, and the isolates comparison is positive externality versus Seq-Compa. In Simultaneous, both players report at the same time; payoffs again depend on whether both reports are zero, the prediction is ambiguous relative to Sequential, and the isolates comparison is strategic uncertainty about the partners report versus Sequential.

^aSequential-Computer; ^bSequential-NoExternality. The isolating comparison for Sequential-NoExternality is with Sequential-Computer, not with the baseline.

3.1. Overview and design

The game used in the experiment is called the mind-cheating gameFootnote ⁷ in which participants choose one color out of five in their minds (see colors in Figure 1). The colors were chosen such that individuals with color vision deficiencies can distinguish them. Then, they draw a color from a deck of cards displayed on their computer screens. The deck of cards contains two cards for each one of the colors. Participants then report whether the color they drew from the deck is the same as their mentally chosen color. If participants want to report that the colors match, they report Yes; otherwise, they report No. Thus, in this game, Yes represents $x_i=1$ and No represents $x_i=0$. For the payoffs, the reward structure assigns payoffs of $v_h=\unicode{x00A3} 2.5$ and $v_l = \unicode{x00A3} 0.3$. In this study, the state of nature is in participants’ minds, so I can only compare distributions of groups based on the known theoretical distribution. However, individual-level cheating cannot be directly observed.Footnote ⁸ A key advantage of the mind-cheating game is that, regardless of the color a participant mentally selects, the probability of a match remains constant and follows a binomial distribution with a known parameter $p=0.2$.

Figure 1.

Colors used in mind game

In the experiment, I use the treatments presented in Table 1. Specifically, in Sequential, $P_1$ reports to $P_2$ whether the colors match or not. Once $P_2$ learns $r_1$, they follow the same sequence of decisions: think a color, draw a color from a deck of cards, and report whether the colors match. In Sequential-Computer, the procedure for $P_1$ is identical. For $P_2$, however, instead of choosing a card mentally, participants select one from a list displayed on their screens and then draw a card from a deck. The computer then truthfully compares the selected color with the drawn color and reports whether they match. Participants know that the computer’s report will always be truthful. In Sequential-NoExternality, decisions are identical to Sequential-Computer, and the variation is that $P_2$’s payoffs only depend on whether their selected color and their drawn color match regardless of $P_1$’s payoffs. Finally, in Simultaneous, both participants think of a color, draw a color, and report at the same time whether the colors match.

While $P_2$ is reporting, I elicit $P_1$’s beliefs about $P_2$’s report. I use a mechanism proposed by Karni (Reference Karni2009) and implemented experimentally first by Mobius et al. (Reference Mobius, Niederle, Niehaus and Rosenblat2011) which allows eliciting probabilities in an incentive-compatible way. Specifically, I use a similar implementation as the one proposed by Coffman (Reference Coffman2011). Participants are asked to predict whether $P_2$ reports Yes or No and are then asked how likely they think their prediction is to be correct. This procedure allows me to elicit the probability of $P_2$ reporting Yes. Participants are told that they do not need to read the instructions about the mechanism or understand it if they do not want to. The concern that complex mechanisms may increase participant attrition is supported by Danz et al. (Reference Danz, Vesterlund and Wilson2022); Danz et al. (Reference Danz, Vesterlund and Wilson2024). To reduce this risk, I use this option. Participants can, however, click a button to view a detailed explanation.Footnote ⁹

Specifically, the elicitation mechanism is based on robots that can guess on behalf of the participants. There are 100 robots, each with integer probability between 1 and 100 of correctly guessing $P_2$’s report. A robot from this interval is drawn randomly, and it can guess on the participant’s behalf with an accuracy level determined by its number. Robot 1 is accurate 1% of the time; robot 2 is accurate 2% of the time, all the way up to the robot that is accurate 100% of the time. The reported likelihood of their guess being correct is used as an “accuracy threshold.” That is, if the robot has an accuracy greater than or equal to the threshold, the robot guesses $r_2$ for $P_1$. If the robot has an accuracy less than the threshold, $P_1$’s guess is submitted. If the guess is correct, whether it is the participant’s or the robot’s, it gives a payoff of $\unicode{x00A3} 0.3$.

3.2. Procedures

I pre-registered the experiment in AEA RCT Registry under the number AEARCTR-0007214. I calculated the power of the target sample size using computer simulations. I used a minimum detectable effect size of 0.15 percentage points from people reporting Yes. The power reached with a sample size of 140 observations by treatment is about 0.8 when simulating 1500 Fisher tests. The experiment was conducted online on Prolific (Palan & Schitter, Reference Palan and Schitter2018) in February 2021. The experiment was programmed in oTree (Chen et al., Reference Chen, Schonger and Wickens2016). A total of 992 people participated in four sessions.Footnote ¹⁰ I did not run the whole experiment in one session to avoid overloading the server and minimize the probability of technical issues. Table E.1 in the Appendix presents the number of observations for people on the role of $P_1$ in each session. The computer program assigned a treatment to each participant. Participants participated only in one treatment, and the game was played only one time. Among the participants, 54.71% identified themselves as male, 44.60% as female, 0.30% as other, and 0.40% preferred not to report it. The average age of participants was 26.25, and 47.28% were students. Participants spent about 7 minutes on average to complete the experiment. In addition to the mind-cheating game earnings and the guessing task, participants received a fixed payment of £1.15 through Prolific. Across the four main sessions, average total compensation was £3.43.