The parameterized complexity of manipulating Top Trading Cycles

Abstract

We study the problem of exchange when agents are endowed with heterogeneous indivisible objects, and there is no money. In this setting, no rule satisfies Pareto-efficiency, individual rationality, and strategy-proofness; there is no consensus in the literature on satisfactory second-best mechanisms. A natural generalization of the ubiquitous Top Trading Cycles (TTC) satisfies the first two properties on the lexicographic domain, rendering it manipulable. We characterize the computational complexity of manipulating this mechanism; we show that it is $\mathbf {W[P]}$-hard by reduction from MONOTONE WEIGHTED CIRCUIT SATISFIABILITY. We provide a matching upper bound for a wide range of preference domains. We further show that manipulation by groups (when parameterized by group size) is $\mathbf {W[P]}$-hard. This provides support for TTC as a second-best mechanism. Lastly, our results are of independent interest to complexity theorists: there are few natural $\mathbf {W[P]}$-complete problems and, as far as we are aware, this is the first such problem arising from the social sciences.

Appendix A

In this section we provide several fully worked examples of the constructions defined in the main text. Previously, in Figs. 5 and 6, we considered a simple circuit with no internal gates, and depicted the corresponding economies $\mathcal {E}'_\mathcal {C}$ and $\mathcal {E}_\mathcal {C}$. In Figs. 8 and 9 we consider a circuit with one internal gate (Fig. 3), and provide the full constructions of the corresponding $\mathcal {E}'_\mathcal {C}$ and $\mathcal {E}_\mathcal {C}$. In Figs. 10 and 11 we detail several steps of TTC for two different circuits and their resulting economy $\mathcal {E}'_\mathcal {C}$ constructions.

Fig. 8

The economy $\mathcal {E}'_\mathcal {C}$ for a circuit with one internal gate

Fig. 9

The economy $\mathcal {E}_\mathcal {C}$ “duplicates” certain elements of $\mathcal {E}'_\mathcal {C}$

Fig. 10

Four steps of TTC for an economy $\mathcal {E}'_\mathcal {C}$

1.1 Appendix A.1: a circuit with one internal gate

Consider the circuit from Fig. 3 equivalent to the formula $x_1\wedge (x_1\vee x_2)$. It is degenerate in a certain sense: it is equivalent to the circuit with a single gate $x_1$ serving as input and output. Thus there is only one satisfying assignment of weight 1, in which $x_1=1$ and $x_2=0$. In Fig. 8, we depict the corresponding economy $\mathcal {E}'_\mathcal {C}$ following the definitions in Sect. 3.1.

By design, there is a beneficial misreport of a particular form corresponding to the satisfying assignment of weight 1. Consider the possible preference relations that agent 1 can report. Recall that their true preference is $R_1= (\alpha ,\beta ,e_\alpha ,e_\beta ,e_k,e_{k-1},\ldots ,e_1)$. If agent 1 reports the truth, then they receive $\{\alpha ,e_\alpha ,e_\beta \}$. If agent 1 reports $R'_1=(x_1^2,\alpha ,\beta ,\square )$ (which represents the satisfying assignment $x_1=1$), then they receive $\{x_1^2,\alpha ,\beta \}$ and are better off. If agent 1 reports $R''_1=(x_2^1,\alpha ,\beta ,\square )$ (which represents a non-satisfying weight 1 assignment $x_2=1$), then they receive $\{x_2^1,\alpha ,e_\beta \}$ and are worse off.

Next, we show the result of the duplication process to construct $\mathcal {E}_\mathcal {C}$ for $k=1$ in Fig. 9. As in Fig. 6, we omit the objects $\alpha$, $\beta$, $\gamma$ , $e_\alpha$, and $e_\beta$ in this diagram for readability.

1.2 Appendix A.2: steps of TTC for economy from Fig. 5

The economy from Fig. 5 is the construction $\mathcal {E}'_\mathcal {C}$ where circuit $\mathcal {C}$ is equivalent to $x_1\wedge x_2$ and $k=1$. Recall that the true preference of agent 1 is $R_1=(\alpha ,\beta ,e_\alpha ,e_\beta ,e_1)$, and that they are allocated $\{\alpha ,e_\alpha ,e_\beta \}$.

In Fig. 10, we show the result of applying TTC when agent 1 misreports to $R'_1=(x_1^*,\alpha ,\beta ,\square )$. For readability, we omit drawing agent 1’s preference except where necessary. In the top left, we depict Step 1 of TTC. Since agent 1 topranks $x_1^*=x^1_1$, we have the cycle $(x_1^*,\gamma _1,e_1)$. All objects that agent 1 owns point to $x_1^*$ as well, but we omit these directed edges. We resolve this cycle. In the top right of the figure, we depict Step 2. We remove objects that were in a cycle in Step 1 (now greyed) and update all agents’ preferences over remaining objects. For example, since $e_1$ is gone, the owner of $\gamma$ now topranks $e_\alpha$. Agent 1 owns $e_\alpha$ which now topranks and points to $\alpha$. Two cycles appear: $(x_2^1,h_3^2)$ and $(\alpha ,\gamma ,e_\alpha )$. We select the former to resolve. In Step 3 (bottom left), we update preferences, and since $h_1^3$ only ranked one object in the previous step, they now toprank their own object. Again there are two cycles: $(h_1^3)$ and $(\alpha ,\gamma ,e_\alpha )$. We select the former to resolve. In Step 4 (bottom right), $(\alpha ,\gamma ,e_\alpha )$ is the only cycle. In Step 5 (not shown), $(y,\beta )$ form a cycle. Thus, we have that agent 1 is allocated $\{x_1^*,\alpha ,e_\beta \}$. This outcome is worse than the outcome if they had told the truth.

Note that in Step 2, removing $h_3^2$ from the graph corresponds to the second predecessor of the $\wedge$-gate $g_3$ taking value 0. Thus, the output $\wedge$-gate $g_3$ takes value 0. The result, by design, is that y cycles with $\beta$, taking away the opportunity for agent 1 to receive $\beta$.

1.3 Appendix A.3: steps of TTC for economy another circuit

In Fig. 11, we show the construction of $\mathcal {E}'_\mathcal {C}$ where $\mathcal {C}$ is the circuit equivalent to $x_1 \vee x_2$ and $k=1$. The true preference of agent 1 is $R_1=(\alpha ,\beta ,e_\alpha ,e_\beta ,e_1)$, and they are allocated $\{\alpha ,e_\alpha ,e_\beta \}$. We show the result of applying TTC when agent 1 misreports to $R'_1=(x_1^*,\alpha ,\beta ,\square )$. Note that this is a satisfying assignment of weight 1 for the circuit. By design, the misreport $R'_1=(x_1^*,\alpha ,\beta ,\square )$ will be beneficial.

Fig. 11

Four steps of TTC for an economy $\mathcal {E}'_\mathcal {C}$ with an $\vee$ output gate

In the top left, we depict Step 1 of TTC. Since agent 1 topranks $x_1^*=x^1_1$, we have the cycle $(x_1^*,\gamma _1,e_1)$. We resolve this cycle. All objects that agent 1 owns point to $x_1^*$ as well, but again we omit these directed edges. In Step 2 (top right), given updated preferences, we have three cycles: $(h^2_3,x_2^1)$, $(y,h_3^1)$, and $(\alpha ,\gamma ,e_\alpha )$. We select the first cycle to resolve. In Step 3 (bottom left), cycles are $(y,h_3^1)$, and $(\alpha ,\gamma ,e_\alpha )$. Again, we select the first cycle to resolve. In Step 4 (bottom right), $(\alpha ,\gamma ,e_\alpha )$ is the only cycle. In Step 5 (not shown), $(\beta ,e_\beta )$ form a cycle. Thus, we have that agent 1 is allocated $\{x_1^*,\alpha ,\beta \}$. This outcome is preferred to the one where they told the truth.