## Abstract

Joint debts are debts that more than one debtor guarantees mutually. They are sometimes interpreted as a version of the tragedy of the commons. However, a simple model of the dynamic tragedy of the commons fails to capture an important feature of joint debts. The authors model joint debts as a joint borrowing limit game and compare the model with the dynamic tragedy of the commons model. Thereby, the authors show the difference in the achievability of efficiency between these two models and present the conditions for efficiency. The conditions for efficiency are as follows: (i) a not too large economic disparity between the players; (ii) not too many players. The joint borrowing limit game has a broad range of applications because the model can be applied to a case where creditors and debtors expect a debt guarantee of someone unilaterally even without an explicit contract of joint debt. For example, this model can be applied to the Eurosystem that led to Greek overborrowing.

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## Notes

- 1.
While this study focuses on Greek overborrowing, De Grauwe (2012) focused on the Greek debt crisis itself. In De Grauwe (2012), there is a sunspot equilibrium in some cases and the default occurs after a realization, which depends on investors’ expectations. De Grauwe (2012) interpreted this as the Greek debt crisis. According to this interpretation, the Greek debt crisis might have been prevented from occurring.

- 2.
The results of this study do not depend on this rule as long as it satisfies \(\sum _{i \in N} c_{i,t} = I({\mathbf {b}}_{t-1})\).

- 3.
\(\text{ G }( N, {\mathbf {e}}, r, u, \beta )\) or “of JBL game \(\text{ G }( N, {\mathbf {e}}, r, u, \beta )\)” will be omitted if there’s no confusion.

- 4.
The results of this paper do not depend on this rule as long as it satisfies \(\sum _{i \in N} c_{i,t} = y_{t-1}\).

- 5.
\(\text{ G }^{DTC}( N, r, u, \beta )\) or “of DTC game \(\text{ G }^{DTC}( N, r, u, \beta )\)” will be omitted if there’s no confusion.

- 6.
See Appendix D for the proof.

- 7.
The utility attained under \(\hat{{\mathbf {s}}}\), \(V({\mathbf {b}})\), is given in Appendix E.

- 8.
The proof that \(g_i(b_i)\) is the policy function of \(\hbox {P}_i ( b_{i,0} )\) and the value function \(V_i^e(b_i)\) are given in Appendix F.

- 9.
- 10.
Hansen and Singleton (1982) reported that \(\sigma\) is between 0.3 and 1.

- 11.
\(C>0.097\) and \(\frac{I_j(b_j)}{I({\mathbf {b}})}\le \frac{1}{11}\).

- 12.
For example, the following expression holds if \(\sigma = 1\):

$$\begin{aligned} \frac{I_i(b_i)}{I({\mathbf {b}})} < \left( \frac{1}{\beta +(1-\beta )n} \right) ^\frac{1}{1-\beta } \iff V(\mathbf{b}) > V_i^e(b_i). \end{aligned}$$\(V(\mathbf{b})\) is the utility under the inefficient MPE and given in Appendix E. \(V_i^e(b_i)\), which is given in Appendix F, is the utility of the efficient MPE (see Appendix I).

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Takahashi, H., Takemoto, T. & Suzuki, A. Can players avoid the tragedy of the commons in a joint debt game?.
*Int J Game Theory* (2020). https://doi.org/10.1007/s00182-020-00722-4

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### Keywords

- Joint debts
- Dynamic game
- Common resource
- Tragedy of the commons
- Greek overborrowing