## Abstract

We propose a theoretical model to explain the usage of time-inconsistent behavior as a strategy to exploit others when reputation and trust have secondary effects on the economic outcome. We consider two agents with time-consistent preferences exploiting common resources. Supposing that an agent is believed to have time-inconsistent preferences with probability *p*, we analyze whether she uses this misinformation when she has the opportunity to use it. Using the model originally provided by Levhari and Mirman (Bell J Econ 11(1):322–334, 1980), we determine the optimal degree of present bias that the agent would like to have while pretending to have time-inconsistent preferences and we provide the range of present-bias parameter under which deceiving is optimal. Moreover, by allowing the constant relative risk aversion class of utility form, we characterize the distinction between pretending to be naive and sophisticated.

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## Notes

- 1.
- 2.
Consider a person with a bad habit of staying up too late. Every morning, he promises to go to bed early, but, at night, he always goes to bed later than he intended. By knowing that he breaks his promises so many times, a rational agent may pre-commit his future behavior. For example, he can say his spouse that he feels tired and sluggish, so that he should go to bed early. If you are married, then you know that your spouse will make you go to bed early either by kindness or by force.

- 3.
For a comprehensive review on psychological determinants of intertemporal preferences, see Urminsky and Zauberman (2015).

- 4.
Note that, even if both players know that they have time-consistent preferences, we might have a subgame perfect equilibrium where the consumption path coincides with the one defined in Proposition 1: The agent 1 continues to play \(g_{S}(x)\) and agent 2 continues to play \(g_{A_{S}}(x)\) as long as no agent deviates from this strategy. If at least one player deviates from this strategy, they both play \(h\left( x\right) =x\) and the resources are exhausted. For a set of payoffs to be supportable in discounted dynamic programming, see Fudenberg and Tirole (1991).

- 5.
The pooling equilibrium that we define resembles the well-known variant of the chain-store game in which there is a small probability

*p*that the monopolist is “tough” and prefers fight rather than cooperate if there is an entry to the market. In the original chain-store game, a monopolist plays against a succession of*K*potential competitors. In each period, one of the potential competitors decide whether or not to compete with the monopolist. If it decides to enter, then the monopolist chooses either to cooperate or to fight. Each potential competitor prefers to stay out rather than entering and being fought, but prefers the most when it enters and the monopolist does not fight. If a competitor enters, the monopolist prefers to cooperate rather than fight, but it prefers the most if there is no entry. In the unique subgame perfect equilibrium of the game, each potential competitor chooses to enter and the monopolists always chooses to cooperate (Selten 1978). Kreps and Wilson (1982) shows that the regular monopolist turns the failure of correct common knowledge about its payoff into an advantage by acting like a tough one and preserves its reputation at least until the horizon gets close. Similarly, we show that agent 1 turns the failure of correct common knowledge about its preferences into an advantage and acts as if he might have problems with self-control. - 6.
With heterogeneous discount factors, we get \(c=\frac{\left( 1-\delta _{1}\alpha \right) \delta _{2}\alpha }{\delta _{2} +\beta \delta _{1}-\alpha \delta _{1}\delta _{2}},\)\(d=\frac{\left( 1-\delta _{2}\alpha \right) \beta \delta _{1}\alpha }{\delta _{2}+\beta \delta _{1} -\alpha \delta _{1}\delta _{2}}\)and \(1-c-d=\frac{\beta \alpha \delta _{1}\delta _{2} }{\delta _{2}+\beta \delta _{1}-\alpha \delta _{1}\delta _{2}}.\)

- 7.
We restrict ourselves to linear strategies to obtain definite results. By relaxing the assumption on output elasticity, one can show numerically that the decision to pretend to have time−inconsistent preferences and the preference between naive and sophisticated behavior may depend on the available resource stock.

- 8.
As we did in Sect. 4, one can solve the model for heterogeneous discount factors. While an MPNE in linear strategies does not exist when agent 1 pretend to be sophisticated, it still exists when both agents act with time-consistent preferences or when agent 1 pretend to be naive. For the naive player, we plot the optimal level of \(\beta \) for multiple cases by freeing modified discount factor of agents one at a time. Our analysis confirms the discussion in Sect. 4 that the optimal level of \(\beta \) depends on the nonlinear interaction of agent 1’s own discount rate and the discount rate of the agent 2.

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## Appendix

### Appendix

### The proof of proposition 2

(a) Follows from Amir and Nannerup (2006).

(b) From part a, one can compute that:

Under Assumption 2, by 7 and 8, we have

Using the linearity of \(g_{N}\left( {}\right) \) and \(g_{A_{N}}(x),\) we find \(g_{N}\left( x\right) =\frac{1-\delta \alpha }{1+\beta -\alpha \delta }x \) and \(g_{A_{N}}(x)=\frac{\beta \left( 1-\delta \alpha \right) }{1+\beta -\alpha \delta }x.\)

(c) Under Assumption 2, by 11 and 12, we have

Using the linearity of \(g_{S}\left( {}\right) \) and \(g_{A_{S}}(x),\) we find \(g_{S}\left( x\right) =\frac{1-\delta \alpha }{1+\beta -\alpha \delta }x \) and \(g_{A_{S}}(x)=\frac{\beta \left( 1-\delta \alpha \right) }{1+\beta -\alpha \delta }x.\)

### The Proof of Corollary 1

It follows from Proposition 2, as we have

### The Proof of Proposition 3

(a) Under case 2 and 3, agent 1’s utility is given by the following:

By guessing that the value function has the form \(A\log x+B,\) we compute

This implies that utility maximizing \(\beta \) solves the following problem :

From the first-order condition, we get \(\beta =\delta \alpha .\) Note that objective function is increasing when \(\beta <\delta \alpha \) and decreasing when \(\beta >\delta \alpha \), i.e., \(\beta =\delta \alpha \) is the unique maximization point.

(b) If the agent cannot choose \(\beta ,\) she gets \(V^{1}\left( x\right) \) if she acts as if she had time-inconsistent preferences and gets \(V_{tc}\left( x\right) \) if she acts truly. Note that \(V^{1}\left( x\right)>V_{tc}\left( x\right) \Longleftrightarrow \log \frac{1}{1+\beta -\delta \alpha }+\delta \alpha \log \beta >\log \frac{1}{2-\delta \alpha }\) . By part (a), we know that left-hand side of the equation is decreasing in \(\beta \) when \(\beta \) is greater than \(\delta \alpha .\) Since they are equal to each other for \(\beta \) equals to 1, the left-hand side is greater than the right-hand side when \(\beta \in [ \delta \alpha ,1) \). Let us consider the case, such that \(( \frac{1}{2-\delta \alpha }) ^{\frac{1}{\delta \alpha }} <\alpha \delta .\) Our result follows from the fact that, for any \(\beta \in [( \frac{1}{2-\delta \alpha }) ^{\frac{1}{\delta \alpha } },\alpha \delta ) ,\) we have \(\log \frac{1}{1+\beta -\delta \alpha } +\delta \alpha \log \beta >\delta \alpha \log \beta \ge \log \frac{1}{2-\delta \alpha }\).

### The Proof of Proposition 4

(a) Under Assumption 3, by 4, we have

By imposing that \(g_{tc}\left( x\right) =a_{tc}x\), we get \(\left( 1-2a_{tc}\right) ^{\sigma }-\delta A^{1-\sigma }\left( 1-a_{tc}\right) =0.\) Since \(h_{tc}\left( 0\right) >0,h_{tc}\left( \frac{1}{2}\right) <0\) and \(h_{tc}\left( c\right) \) is continuous in *c* implies that there exist \(a_{tc} \)\(\in \left( 0,\frac{1}{2}\right) \), such that \(h_{tc}\left( a_{tc}\right) =0.\) By 5, we get:

(b) Under Assumption 3, by 7 and 16, we have

By imposing that \(g_{N}\left( x\right) =a_{n}x,\) and \(g_{A_{N}}(x)\) is linear in *x*, we get \(g_{A_{N}}(x)=\left( 1-a_{n}\left( 1+B^{1/\sigma }\right) \right) x\) and

Consider function \(h_{N}\left( {}\right) \). In equilibrium, we must have \(x-g_{N}\left( x\right) -g_{A_{N}}(x)>0\), i.e., \(a_{n}<\frac{1}{1+B^{1/\sigma }}.\)Since \(h_{N}\left( a_{tc}\right) =Ba_{tc}^{\sigma }-\delta A^{1-\sigma }\left( 1-a_{tc}\right) =\frac{\delta A^{1-\sigma }\left( \beta -1\right) a_{tc}}{1-\delta A^{1-\sigma }\left( 1-2a_{tc}\right) ^{1-\sigma }}<0,\)\(h_{N}^{\prime }\left( c\right) >0,\) and by assumption of the proposition \(h_{N}\left( \frac{1}{1+B^{1/\sigma }}\right) >0,\) there exists unique \(a_{n}\)\(\in \left( a_{tc},\frac{1}{1+B^{1/\sigma }}\right) \), such that \(h_{N}\left( a_{n}\right) =0.\)

(c) Under Assumption 3, by 11, we have

By imposing that \(g_{S}\left( x\right) =a_{s}x,\) and \(g_{A_{S}}(x)\) is linear in *x*, we get \(g_{A_{S}}(x)=\beta a_{s}x\) and

In equilibrium, we must have \(x-g_{S}\left( x\right) -g_{A_{S}}(x)>0\) i.e. \(a_{s}<\frac{1}{1+\beta }.\)Consider function \(h_{S}\left( {}\right) \). Since \(h_{S}\left( a_{tc}\right) >h_{tc}\left( a_{tc}\right) =0,\)\(h_{S}\left( \frac{1}{1+\beta }\right) <0\) and \(h_{S}\left( c\right) \) is continuous in c, there exists \(a_{s}\)\(\in \left( a_{tc},\frac{1}{1+\beta }\right) \), such that \(h_{S}\left( a_{s}\right) =0.\)

### The proof of corollary 2

Since the equilibrium strategy of a naive and a sophisticated agent does not necessarily coincide, we have two different cases to consider. Let us consider the strategic interaction with a naive agent first. We have

where \(c_{n}x=g_{N}\left( x\right) +g_{A_{N}}(x)=a_{n}x+\left( 1-a_{n}\left( 1+B^{1/\sigma }\right) \right) x\).Since \(\left( 1-2a_{tc}\right) ^{\sigma }-\delta A^{1-\sigma }\left( 1-a_{tc}\right) =0\), we have the relation that:

By Proposition 4b, we conclude \(g_{N}\left( x\right) +g_{A_{N}}(x)>2g_{tc}\left( x\right) .\)Similarly, we have

where \(c_{s}x=g_{S}\left( x\right) +g_{A_{S}}(x).\)This implies that:

By Proposition 4c, we conclude \(g_{S}\left( x\right) +g_{A_{S}}(x)>2g_{tc}\left( x\right) .\)

### The pooling equilibrium and optimal value of discount factor when there is an uncertainty about agent 1’s discount factor

Suppose that agent 1’s true discount factor is \(\delta _{1}\), while agent 2 believes that her discount factor is \(\hat{\delta }_{1}\) with probability *p*. Consider the games with discount rate pairs \(\left( \hat{\delta }_{1} ,\delta _{2}\right) \) and \(\left( \delta _{1},\delta _{2}\right) .\)Since both agents have time-consistent preferences, MPNE of the games can be found by following the steps that we define in Sect. 2.1. Let us denote MPNE by \(\hat{g}_{tc}^{1}(x),\hat{g}_{tc}^{2}(x)\) and value functions by \(\hat{V}_{tc}^{1}(x),\hat{V}_{tc}^{2}(x)\) for the game with discount rate \((\hat{\delta }_{1},\delta _{2}).\) Similarly, denote MPNE of the game with discount rate \(( \delta _{1},\delta _{2})\) by \(g_{tc}^{1}(x),g_{tc}^{2}(x)\) and the corresponding value functions by \(V_{tc}^{1}(x),V_{tc}^{2}(x).\) We can define the pooling equilibrium as below:

If \(\hat{V}_{tc}^{1}(x)\ge V_{tc}^{1}(x)\) and \(\hat{V}_{tc}^{i}(x)\ge IC (x) \), where \(i\in \left\{ 1,2\right\} ,\) there is a perfect Bayesian equilibrium \(\left( \left( \tilde{s}^{1}(h,\theta ),\tilde{s} ^{2}\left( h\right) \right) ,\left( \mu \left( h\right) \right) \right) \), such that

Under Assumption 2, by Levhari and Mirman (1980), we have:

Agent 1 finds the optimal value of \(\hat{\delta }_{1}\) by solving the trade-off between her consumption rate and combined investment rate governed by the following maximization problem:

where *c* and *d* represent the equilibrium consumption rates of the agents, i.e., \(c=\frac{\hat{g}_{tc}^{1}(x)}{x}\) and \(d=\frac{\hat{g}_{tc}^{2}(x)}{x}\). From the first-order condition, we get \(\hat{\delta }_{1}=\frac{\alpha \delta _{1}\delta _{2}}{1-\alpha \delta _{1}+\alpha ^{2}\delta _{1}\delta _{2}}.\)Note that objective function is increasing when \(\hat{\delta }_{1}<\frac{\alpha \delta _{1}\delta _{2}}{1-\alpha \delta _{1}+\alpha ^{2}\delta _{1}\delta _{2}}\) and decreasing when \(\hat{\delta }_{1}>\frac{\alpha \delta _{1}\delta _{2}}{1-\alpha \delta _{1}+\alpha ^{2}\delta _{1}\delta _{2}}\), i.e., \(\hat{\delta } _{1}=\frac{\alpha \delta _{1}\delta _{2}}{1-\alpha \delta _{1}+\alpha ^{2}\delta _{1}\delta _{2}}\) is the unique maximization point.

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Turan, A.R. Intentional time inconsistency.
*Theory Decis* **86, **41–64 (2019). https://doi.org/10.1007/s11238-018-9671-y

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### Keywords

- Time-inconsistent preferences
- Hyperbolic discounting
- Dynamic game
- Common property resources
- Perfect Bayesian equilibrium