Why DeDeal Works

Intro

When DeDeal is modeled using "A SIMPLE MODEL OF INEQUITY AVERSION" from the research paper referenced below, we can show that fraud does not occur between players with tendencies of "inequity aversion."

“Inequity aversion" is the tendency to resist unfair outcomes (i.e. they are willing to forego some material rewards in order to obtain a fairer outcome).

There is substantial evidence to suggest that "inequity aversion" influences the behavior of many people, seen through the results of both economic experiments such as the ultimatum game and real-world empirical results in corporate wage setting.

Reference：Fehr, Ernst, and Klaus M. Schmidt. “A Theory of Fairness, Competition, and Cooperation.” The Quarterly Journal of Economics, vol. 114, no. 3, 1999, pp. 817–68. JSTOR.

Modeling

General Model

“A SIMPLE MODEL OF INEQUITY AVERSION” for players i and j is as follows

$U_i(x) = x_i - \alpha_i \times \max(x_j - x_i, 0) - \beta_i \times \max(x_i - x_j, 0), i \ne j$

$U_i(x)$ : Player i's utility function

$x_i$: Player i's monetary utility

$x_j$: Player j's monetary utility

$\alpha_i$: Player i's utility loss multiplier due to inequality against their favor

$\beta_i$: Player i's utility loss multiplier due to inequality in their favor

Seller fraud pattern

Applying this model to the situation where the seller commits fraud and fails to ship the item at price P, the options for the buyer and seller are as follows

・Phase 0

Choice of whether or not the seller commits fraud

・Phase 1

Choice of whether or not the buyer presses the sanction button in the case of fraudulent non-shipment of item by the seller

$U_b(P)$: Buyer's utility function

$P$: Price of the product

$x_b$: Buyer's monetary utility

$x_s$: Seller's monetary utility

$α_b$: Buyer's utility loss multiplier due to inequality against their favor

$β_b$: Buyer's utility loss multiplier due to inequality in his favor

$R_b$: Buyer's utility loss due to the risk that the buyer deposit to press the sanction button is lost due to the seller sanctioning back

<Option 1-1>

Buyer's utility function if the buyer does not press the sanction button:

\begin{equation} U_b(P) = -P - \alpha_b \cdot (P-(-P)) - \beta_b \cdot (0) = -P - \alpha_b \cdot 2P \end{equation}

<Option 1-2>

Buyer's utility function if the buyer presses the sanction button:

$U_b(P) = -P - \alpha_b \times (0) - \beta_b \times (0) - R_b = -P - R_b$

($Rb$ is $2P$ if the seller presses the sanction button in Phase 2 and $0$ if not.)

・Phase 2

Choice of whether or not the seller presses the sanction button when the buyer chooses to sanction

$U_s(P)$: Seller's utility function

$P$: Price of the item

$x_s$: Seller's monetary utility

$x_b$: Buyer's monetary utility

$α_s$: Seller's utility loss multiplier due to inequality against their favor

$β_s$: Seller's utility loss multiplier due to inequality in their favor (this value is $0$ because the seller is committing fraud)

<Option 2-1>

Seller's utility function if the seller does not press the sanction button:

$U_s(P) = -P - \alpha_s \times (0) - \beta_s \times (0)$

<Option 2-2>

Seller's utility function if the seller presses the sanction button:

$U_s(P) = -P - \alpha_s \times (0) - 0 \times (-P - (-3P)) - R_s = -P - R_s$

($Rs$ is $2P$ if the buyer sanctions back afterwards and $0$ if not)

Now, assuming that the buyer chooses <Option 2> in phase 1, we can expect that in phase 2 the seller will compare the $U_s(P)$ values associated with <options 2-1> and <options 2-2> and opt for option 3.

If <Option 2-1> is chosen in Phase 2, then since "Rb" is 0 in Phase 1, we can expect the buyer to compare the values of Ub(P) when choosing <Option 1-1> and <Option 1-2>, and choose <Option 1-2>.

In other words, if the seller chooses to commit fraud in Phase 0, the expected final utility of each option is as follows, which means that if the buyer is an "inequity aversive" player, the seller should not commit fraud.

$U_s(P)=-P$

$U_b(P)=-P$