Arts and craftiness: an economic analysis of art heists

Abstract

We study the incentives that museums face in determining how much resources to invest in the protection of their artwork from theft. We present and analyze a game-theoretic model of art heists that accounts for the strategic interactions between museums’ and art thieves’ decisions and that incorporates several key features of the black market for stolen art. We find that the equilibrium level of security museums choose need not be monotonic in the true market value or the black market value of artwork, i.e., increasing the value of an art piece—whether it is the true market value or the black market value—does not necessarily lead museums to invest more in protecting their artwork. The effects of parameter changes in the model that reflect a shift of public policy depend critically on what type of policy change is considered. For instance, an increase in the penalty imposed for committing art theft cannot raise the amount of theft in equilibrium and could in fact lead museums to increase their level of security. On the other hand, investing more resources on law enforcement agencies so that they are better able to solve art crimes can actually increase the amount of theft in equilibrium by causing museums to spend less on security.

Appendix

Proof of Proposition 4

It is easy to find parameter values such that decreasing $C_{\mathrm{h}}$ causes the museum to switch from low security to high security. To prove the second part of the proposition, consider first Case 1, where $\alpha \le 1-\delta \left( V\right)$. Suppose given the initial value of $C_{\mathrm{h}}$, the museum chooses the high level of security. Inspection of the conditions stated in Proposition 1 shows that there are only two ways for this to occur:

• $\delta \left( V\right) V\ge \frac{d_{\mathrm{h}}+\gamma _{\mathrm{h}}p}{1-\gamma _{\mathrm{h}}}$ and $\delta \left( V\right) V\ge \frac{C_{\mathrm{h}}-C_{\mathrm{l}}}{\gamma _{\mathrm{h}}-\gamma _{\mathrm{l}}}$

• $\frac{d_{\mathrm{l}}+\gamma _{\mathrm{l}}p}{1-\gamma _{\mathrm{l}}}\le \delta \left( V\right) V\le \frac{d_{\mathrm{h}}+\gamma _{\mathrm{h}}p}{1-\gamma _{\mathrm{h}}}$ and $\delta \left( V\right) V\ge \frac{C_{\mathrm{h}}-C_{\mathrm{l}}}{1-\gamma _{\mathrm{l}}}$.

When the value of $C_{\mathrm{h}}$ decreases, both conditions above will still hold, i.e., the museum would still prefer the high level of security. Therefore, the museum would not switch from high security to low security in this case.

Now, consider Case 2, when $\alpha \ge 1-\delta \left( V\right)$. It is obvious from Fig. 5 that a decrease in $C_{\mathrm{h}}$ makes the regions in which the museum chooses the high security level bigger. Consequently, we could never have a situation in which the museum switches from the high security level to the low security level. $\square$

Proof of Proposition 5

The first part of Proposition 5 is straightforward to demonstrate. For the second part, first note that changing the value of $C_{\mathrm{h}}$ has no effect on the thief’s equilibrium strategy since $C_{\mathrm{h}}$ does not enter into the thief’s payoffs. Now, suppose to the contrary we observe no heist given $C_{\mathrm{h}}=c_{2}$ and see a heist given $C_{\mathrm{h}}=c_{1}$. Whether we are in Case 1 or 2, the only way this could happen—given that changing $C_{\mathrm{h}}$ does not alter the thief’s equilibrium strategy—is for the SPE of the game to switch from $\left( h\text {, Steal, No Steal}\right)$ given $C_{\mathrm{h}}=c_{2}$ to $\left( l\text {, Steal, No Steal}\right)$ given $C_{\mathrm{h}}=c_{1}$. However, Proposition 4 tells us that this cannot happen; therefore, the second part of Proposition 5 must be true. $\square$

Proof of Proposition 7

To see this, note that the conditions given in Propositions 1 and 3 imply the following, whether we are in Case 1 ($\alpha \le 1-\delta \left( V\right)$) or Case 2 ($\alpha \ge 1-\delta \left( V\right)$).

• If $\left( h\text {, Steal, Steal}\right)$ is a SPE given $p=p_{1}$, then $\left( h\text {, Steal, Steal}\right)$, $\left( l\text {, Not Steal, Not Steal}\right)$, $\left( h\text {, Steal, Not Steal}\right)$, or $\left( l\text {, Steal, Not Steal}\right)$ is a SPE given $p=p_{2}$.

• If $\left( l\text {, Steal, Steal}\right)$ is a SPE given $p=p_{1}$, then $\left( l\text {, Steal, Steal}\right)$, $\left( l\text {, Not Steal, Not Steal}\right)$, $\left( h\text {, Steal, Not Steal}\right)$, or $\left( l\text {, Steal, Not Steal}\right)$ is a SPE given $p=p_{2}$.

• If $\left( l\text {, Not Steal, Not Steal}\right)$ is a SPE given $p=p_{1}$, then $\left( l\text {, Not Steal, Not Steal}\right)$ is a SPE given $p=p_{2}$.

• If $\left( h\text {, Steal, Not Steal}\right)$ is a SPE given $p=p_{1}$, then $\left( h\text {, Steal, Not Steal}\right)$ or $\left( l\text {, Not Steal, Not Steal}\right)$ is a SPE given $p=p_{2}$.

• If $\left( l\text {, Steal, Not Steal}\right)$ is a SPE given $p=p_{1}$, then $\left( l\text {, Steal, Not Steal}\right)$ or $\left( l\text {, Not Steal, Not Steal}\right)$ is a SPE given $p=p_{2}$.

These derivations tell us that it is not possible to have a situation in which no heist occurs in equilibrium given $p=p_{1}$, while a heist occurs in equilibrium given $p=p_{2}$. $\square$