Pricing Anonymity

Abstract

In electronic anonymity markets a taker seeks a specified number of market makers in order to anonymize a transaction or activity. This process requires both coalition formation, in order to create an anonymity set among the taker and makers, and the derivation of the fee that the taker pays each maker. The process has a novel property in that the taker pays for anonymity but anonymity is created for both the taker and the makers. Using the Shapley value for nontransferable utility cooperative games, we characterize the formation of the anonymity set and the fee for any arbitrary number of makers selected by the taker.

Appendices

Appendix

A Shapley Value Derivation for the 3-Player Example

Following the formula given in Eq. (2), the Shapley value for the taker, $t$, with the makers as players 1 and 2, is

$$ \begin{aligned} \varphi _t(\omega , \varvec{\lambda }) =&\frac{1}{3} \bigl (\omega (N)-\omega (\{1, 2\})\bigr )\,+&\text {(grand coalition)} \nonumber \\&\frac{1}{6} \bigl (\omega (\{t, 1\})-\omega (\{1\})\bigr )\,+&\text {(taker}\, \& \, \text {maker 1)} \nonumber \\&\frac{1}{6} \bigl (\omega (\{t, 2\})-\omega (\{2\})\bigr )\,+&\text {(taker}\, \& \, \text {maker 2)} \nonumber \\&\frac{1}{3} \bigl (\omega (\{t\})-\omega (\emptyset )\bigr ).&\text {(taker alone)} \end{aligned}$$

(32)

Substituting in the worth function values, Eqs. 10–15,

(by convention, $\omega (\emptyset )=0$):

$$\begin{aligned} \varphi _t(\omega , \varvec{\lambda })&= \frac{1}{3}\left( \frac{2}{3}D+ \frac{4}{3}d\right) + \frac{1}{6}\left( \frac{1}{2}D+ \frac{1}{2}d\right) + \frac{1}{6}\left( \frac{1}{2}D+ \frac{1}{2}d\right) + \frac{1}{3}\left( \delta D-F\right) . \end{aligned}$$

(33)

Aggregating terms,

(34)

and simplifying:

$$\begin{aligned}&= \frac{14}{36}D+ \frac{22}{36}d+ \frac{1}{3}\left( \delta D-F\right) . \end{aligned}$$

(35)

Using Eq. (2) to calculate the Shapley value for player 1, who is a maker:

(36)

Substituting in the worth function values:

$$\begin{aligned} \varphi _1(\omega , \varvec{\lambda })&= \frac{1}{3}\left( \frac{2}{3}D+ \frac{4}{3}d- \frac{1}{2}D- \frac{1}{2}d\right) + \frac{1}{6}\left( \frac{1}{2}D+ \frac{1}{2}d- \left( \delta D-F\right) \right) . \end{aligned}$$

(37)

Aggregating terms and simplifying:

$$\begin{aligned} \varphi _1(\omega , \varvec{\lambda })&= \left( \frac{2}{9}-\frac{1}{6}+\frac{1}{12}\right) D+ \left( \frac{4}{9}-\frac{1}{6}+\frac{1}{12}\right) d- \frac{1}{6}\left( \delta D-F\right) \end{aligned}$$

(38)

$$\begin{aligned}&= \frac{5}{36}D+ \frac{13}{36}d- \frac{1}{6}\left( \delta D-F\right) . \end{aligned}$$

(39)

By the symmetry property of the Shapley value, $\varphi _2(\omega , \varvec{\lambda }) = \varphi _1(\omega , \varvec{\lambda })$.    $\square$

B Proof of Theorem 2

The proof consists of three parts.

1.1 B.1 Shapley Value for the Taker

Proof

From Eq. (2), the coefficient on $\omega (\{t\})-\omega (\emptyset ) = \delta D-F$ in the Shapley value is $1/N= 1/(m+1)$. This is the first term in Eq. (28).

Given $m$ makers, there are ${m\atopwithdelims ()n} = \frac{m!}{n!\,(m-n)!}$ combinations of coalitions that can be expressed as $S=\{t, n\}$. Note also that $N= m+1$. From Eq. (2), the coefficient on each coalition $\{t, n\}$ in the Shapley value is

$$\begin{aligned} \frac{(|S|-1)!\,(N-|S|)!}{N!}&= \frac{\bigl ((n+1)-1\bigr )!\,\bigl ((m+1)-(n+1)\bigr )!}{(m+1)!} = \frac{n!\,(m-n)!}{(m+1)\,m!}. \end{aligned}$$

(40)

For each coalition, $\{t, n\}$, the marginal contribution for the taker in the formula for $\varphi _t$ is $\bigl (\omega (\{t, n\})-\omega (\{n\})\bigr )=\omega (\{t, n\})$. In aggregate, the partial sum in the Shapley value for a specific $n$ is the product of the following three terms: (i) the number of $\{t, n\}$ coalitions, (ii) the Shapley coefficient that is common to each $\{t, n\}$ coalition, (40), and (iii) $\bigl (\omega (\{t, n\})-\omega (\{n\})\bigr )=\omega (\{t, n\})$, from Eq. (27):

$$\begin{aligned}&\frac{m!}{n!\,(m-n)!} \times \frac{n!\,(m-n)!}{(m+1)\,m!} \times \omega (\{t, n\} = \frac{1}{m+1} \left( \frac{n}{n+1}D+ \frac{n^2}{n+1}d\right) .&\end{aligned}$$

(41)

Summing this over all possible $n= 1, \dots , m$ yields the final two terms in Eq. (28). This completes the derivation of the Shapley value for the taker, $\varphi _t/\lambda _t$, given that $\lambda _t= 1$.

1.2 B.2 Shapley Value for the Makers

In deriving the Shapley value for maker i, $\varphi _i$, note that for any coalition, $\hat{S}$, where . This simplifies the remaining steps for calculating the Shapley value for i to only those worth functions whose coalitions include a taker, $t$, as a member; i. e., $\omega (\{t, n\})$.

Proof

For any maker, i, there is one and only one $\{t, i\}$ coalition. The coefficient on this coalition in the Shapley value is

$$\begin{aligned} \frac{(2-1)!\,\bigl ((m+1)-2\bigr )!}{(m+1)!}&= \frac{(m-1)!}{(m+1)!} = \frac{1}{m\cdot (m+1)}. \end{aligned}$$

(42)

As $\omega (\{t, i\})=\nicefrac {1}{2}D+ \nicefrac {1}{2}d$ and $\omega (\{t, i\}\setminus \{i\}) = \delta D-F$, the part of the calculation of $\varphi _i$ that corresponds to the marginal contribution of i to $\{t, i\}$ is:

$$\begin{aligned}&\frac{1}{m\cdot (m+1)} \Bigl (\omega (\{t, i\})-\omega (\{t, i\}\setminus \{i\})\Bigr ) = \frac{1}{m\cdot (m+1)} \left( \frac{1}{2}D+ \frac{1}{2}d- (\delta D-F)\right) .&\end{aligned}$$

(43)

From Eq. (2) the calculation of the Shapley value is now:

(44)

The remainder of the coalitions where $i \in \{t, n\}$ require $n\ge 2$. For a given $n$, the number of coalitions for which maker i is a member, $i \in \{t, n\}$, is

(45)

Given $n$, the coefficient in the Shapley value for the marginal contribution, $\omega (\{t, n\})-\omega (\{t, n\}\setminus \{i\})$, of maker i is

(46)

To calculate $\omega (\{t, n\})-\omega (\{t, n\}\setminus \{i\})$, use Eq. (27) and observe that

(47)

Consequently, i’s marginal contribution to the coalition $\{t, n\}$ is

(48)

Hence, when $n\ge 2$, for a given value of $n$ the partial sum within the Shapley value corresponding to $\{t, n\}$ such that $i \in \{t, n\}$ is the product:

(49)

Summing this over all possible $n= 2, \dots , m$ yields:

(50)

To derive $\varphi _i$, combine this with the Shapley value term for $\{t, i\}$, Eq. (43):

(51)

Aggregating terms:

(52)

This completes the derivation of the Shapley value for a maker, $\varphi _i/\lambda _i$, given $\lambda _i = 1$.

1.3 B.3 Feasibility Check

The final step requires verification that $\varphi _i/\lambda _i$ is feasible for $V(N)$. This is facilitated using the expressions of $\varphi _t/\lambda _t$ and $\varphi _i/\lambda _i$ in terms of harmonic numbers, as given in Eqs. (56) and (57) in Appendix C.

Proof

Recall that the fee, $f$, given in Eq. (31) was derived from the feasibility condition for $\varphi _t/\lambda _t$ when $\lambda _t= 1$: $\varphi _t\le \frac{m}{m+1}D-mf$, yielding $f= \frac{1}{m+1}D-\frac{1}{m}\varphi _t$. The feasibility condition for $\varphi _i/\lambda _i$ when $\lambda _i=1$ is

$$\begin{aligned} \varphi _i \le \frac{m}{m+1}d+f&= \frac{m}{m+1}d+ \frac{1}{m+1}D- \frac{1}{m}\varphi _t. \end{aligned}$$

(53)

Setting the two sides equal and substituting in the value for $\varphi _t$, Eq. (56):

(54)

which reduces to

(55)

This is exactly the right side of Eq. (57). Hence the condition holds under the theorem.

C Alternative Form of Theorem 2 Using Harmonic Numbers

Let $H_{m}$ denote the $m$-th harmonic number, i.e., $H_{m}=\sum _{n=1}^m\frac{1}{n}$. Using this shorthand, Eq. (28) of Theorem 2 can be rewritten as,

$$\begin{aligned} \frac{\varphi _t(\omega , \varvec{\lambda })}{\lambda _t}&= \frac{\delta D-F+ D\cdot (m- H_{m+1} +1) + d\cdot \left( H_{m+1}+\left( \frac{m}{2}-1\right) \left( m+1\right) \right) }{m+1}; \end{aligned}$$

(56)

and Eq. (29) becomes:

(57)