Experimental Design Details
In our game, a reporter (Player 1) is paired with an audience (Player 2). Player 1 rolls a physical six-sided die and reports the outcome of the die roll. The report is known to Player 2, who, moreover, does not observe the outcome. This report determines Player 1's payoff and may also affect Player 2’s payoff, with all payoff rules being common knowledge. The first treatment arm investigates whether the externality imposed on a regular audience affects one’s reporting behavior. We manipulate the externality by adjusting the audience’s payoff function and implement three treatments: EXT^- (negative externality), EXT^0 (zero externality) and EXT^+ (positive externality).
Our first hypothesis stems from a unique aspect of Dufwenberg & Dufwenberg (2018) (D&D) theory that Player 2’s payoff does not factor into Player 1’s utility function. This suggests that Player 1’s decisions are independent of Player 2’s potential gains or losses.
Hypothesis 1: There is no difference between the distribution of reports in treatments EXT^-, EXT^0, and EXT^+.
The second treatment arm explores whether Player 1’s behavior is independent of the experimenter’s ability to observe the die-roll once a regular Player 2 is involved. We introduce three additional treatments wherein the experimenter observes die-roll: OBS^-, OBS^0 and OBS^+. In these treatments, Player 1 rolls a six-sided die under the observation of the experimenter, who then records the die-roll. It is common knowledge that Player 2 will never learn the actual die-roll outcome and cannot link the report to a specific player. Specifically, the payoff structure in OBS^- is identical to that in EXT^-, with analogous parallels between OBS^0 and EXT^0, as well as OBS^+ and EXT^+.
In hypothesis 2, we test if experimenter’s observability of the initial outcome influences Player 1’s behavior.
Hypothesis 2: (i) The distribution of reports in EXT^- and OBS^- do not differ; (ii) The distribution of reports in EXT^0 and OBS^0 do not differ; and (iii) The distribution of reports in EXT^+ and OBS^+ do not differ.
Our subsequent tests are contingent on the outcome of Hypothesis 2. If Hypothesis 2 is rejected, our next step would be to examine whether the sailing-to-ceiling equilibrium derived from the extension of D&D's Perceived Cheating Aversion theory predicts behavior patterns across all treatments in Hypothesis 3 to 5. Conversely, if Hypothesis 2 is not rejected, this scenario sets the stage for comparing the theoretical predictions of D&D vs. Gneezy, Kajackaite & Sobel (2018) (GKS), and Khalmetski & Sliwka (2019) (K&S), using the data from the observed treatments OBS^- , OBS^0, and OBS^+.
If Hypothesis 2 is rejected, we test Hypothesis 3-4:
Hypothesis 3: The distribution of Player 1’s report in unobserved games first-order stochastically dominates that in observed games: F_(EXT^- ) (y)≤F_(OBS^- ) (y), F_(EXT^0 ) (y)≤F_(OBS^0 ) (y), and F_(EXT^+ ) (y)≤F_(OBS^+ ) (y). Equality is achieved only at y=5.
Hypothesis 4: In the treatments OBS^-，OBS^0，and OBS^+: if the true outcome x∈{1,2,…,5}, then possible reports y are within {x,x+1,…,5}; if x=6, then y ranges across {1,2,…,6}.
If Hypothesis 2 is not rejected, we test the predictions in D&D vs. GKS+K&S.
Hypothesis 5 (D&D vs. GKS+K&S) In treatments OBS^-, OBS^0, and OBS^+:
D&D Prediction: A report y∈{1,2,3,4,5} may originate from a Player 1 who observes x∈{1,2,…,y}∪{6}; a report of y=6 occurs solely if Player 1 observes x=6.
GKS+K&S Prediction: (i) if a Player 1 opts to lie by reporting k∈{1,2,3,4,5}, then no player who actually observes x=k will choose to lie; (ii) there exists a threshold value k ̅ (1<k ̅<5), such that a Player 1 observing x∈{1,…,k ̅-1}∪{6} will report y∈{k ̅,…,5}, and a Player 1 observing x∈{k ̅,…,5} will truthfully report y=x.