## Abstract

We design incentive schemes for portfolio managers that screen low-skill managers: only the best portfolio managers, in terms of expected payoffs, agree to participate in a single-period investment. The results hold in general financial markets, where uninformed investors face managers of different capabilities, and can only observe their one-stage realized returns.

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## Notes

A liquidation boundary guarantees that investors are informed of the investment’s status once it reaches a predetermined low threshold; this allows them to liquidate their investment at the given threshold.

We should emphasize that deposits need not be actually transferred. The notion of transferable deposits could be applied through a simple reimbursement rule by the investor.

A binary option is an investment tool where the payoff is either a fixed monetary amount, or nothing at all; this interaction between managers resembles to some extent the more recent trend of studies concerning optimal contracting jointly with asset prices. See e.g., Qiu (2017), Buffa et al. (2014), Malamud and Petrov (2014) and Garleanu and Pedersen (2018).

Fortune, “Why Buffett’s Million-Dollar Bet Against Hedge Funds Was a Slam Dunk” on May 11, 2016.

The complexity of the given problem will become more evident after our impossibility result in Sect. 3.2.

The manipulative abilities of probabilistic forecasters, with no superior forecasting abilities, were proven to exist in many papers in the expert-testing literature; see e.g., Foster and Vohra (1998), Fudenberg and Levine (1999), Lehrer (2001), Sandroni et al. (2003), Sandroni (2003), Shmaya (2008) and Olszewski and Sandroni (2008).

To clarify, we assume that one manager is high-skilled, the other is low-skilled, and the investor’s main goal is to distinguish between the managers. Our analysis does not account for other scenarios where, for example, both managers are low-skilled and the investor prefers not to invest at all.

See e.g., Stracca (2006) for a survey of the screening problem and previously mentioned impossibility results.

The return on an investment is its realized value, whereas the profit relates to the surplus. For example, a \(5\%\) profit, turning \(\$100\) into \(\$105\), is referred to as a 1.05 return on the investment.

Though this goes beyond the scope of our model, note that Theorem 1 is easily extended to a broader set of cost functions. For example, if the opportunity costs are positively fixed and uniform across managers (i.e., independent of either \(v_i\) or

*i*), then screening is still possible for a range of parameters \(\lambda\) and*C*. Moreover, as long as the opportunity costs are positive and the ratio between the opportunity cost of manager 1 and that of manager 2 is bounded from above by \(\tfrac{2v_1 - \varepsilon }{2v_2 - \varepsilon }\) for the stated \(\epsilon\) of \(V_{\epsilon }\), one can adjust the TD scheme to ensure a screening mechanism.To simplify the computations and without loss of generality, we assume that

*w*is uniform among active managers.See the broad discussion regarding this issue in Sect. 4.3.

The trivial case of a single active manager \(j >i\) is impossible, since

*C*is bounded by \(\lambda w \alpha _i\).

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## Acknowledgements

The authors wish to thank Eddie Dekel, David Gilo, Zvika Neeman, Dotan Persitz, David Schmeidler, Eilon Solan, Avi Wohl, two anonymous reviewers and the associate editor for their valuable comments, as well as the participants of Games 2016, the 2017 Research Workshop in honor of Ehud Kalai, the Tel-Aviv University Game Theory and Mathematical Economics Research Seminar, the Strategy Seminar of the Tel-Aviv University Business School. Lagziel and Lehrer acknowledge the support of the Israel Science Foundation, Grant #963/15. An earlier version of this paper was titled “The Tried-Stone Scheme & a Million Dollar Bet”. Both authors have no relevant or material financial interests that relate to the research described in this paper.

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## Appendix

### Appendix

### 1.1 Proof of Theorem 1

###
*Proof*

Given that both managers are active, the linearity of \(f_i\) w.r.t. the return of manager *i* implies that the dominant-strategy equilibrium is \(\sigma =(Y_1,Y_2)\). Hence, the expected payoffs of the two managers are

where the first inequality follows from the lower bound on \(\tfrac{C}{\lambda w}\) and the fact that \(1 - v_2 + \tfrac{v_2 - v_{1}}{V_{\epsilon }}<0\); for the second inequality, either the term \(1 - v_1 + \frac{v_1 - v_2}{V_{\epsilon }}\) is non-negative and the inequality is evident, or the term is negative and we use the upper bound on \(\tfrac{C}{\lambda w}\). Thus, manager 2 cannot gain from being active along with manager 1, whereas the dominant strategy of manager 1 dictates that he should always remain active. We conclude that manager 1 is the only active manager and he invests the entire amount of 2*w* in \(Y_1\), as needed. \(\square\)

### 1.2 Proof of Proposition 1

###
*Proof*

Fix a feasible scheme *f* with a deposit \(C\geqslant 0\). Assuming both managers are active, their expected payoffs are

Denote \(C_+ = \tfrac{(1-p)f(a,0)+p f(0,a)}{(1-p)a}\) and \(C_{-} = \tfrac{p f(a,0)+(1-p) f(0,a)}{pa}\). A straightforward examination shows that \(p>0.5\) yields \(C_{-} \leqslant C_{+}\). One can verify that a deposit \(C\leqslant C_{+}\) ensures that \(\pi _2\geqslant 0\), thus manager 2 would become active. Otherwise, a deposit \(C>C_{+}\) produces negative expected payoffs for both managers in case both are active. Thus, there exist two equilibria, where in each a different manager is active, and this concludes the proof. \(\square\)

### 1.3 Proof of Lemma 1

###
*Proof*

Assume, by contradiction, that there exists an equilibrium \(\sigma\) where manager *j* participates, but manager \(i<j\) does not. Without loss of generality, assume that manager *j* is the lowest-skill active manager (w.r.t. expected optimal returns). The GS is linearly increasing in an active manager’s return and linearly decreasing in the other active-managers’ average return. Thus, if manager *i* decides to participate, his expected payoff will be higher than manager *j*’s payoff for two reasons. First, manager *i* produces excess expected return relative to manager *j*, as \(v_i>v_j\). Second, manager *i* also gains from the reduced benchmark, which consists of the returns of other active managers (including manager *j*), compared to manager *j*’s benchmark in the assumed equilibrium \(\sigma\) (which does not include manager *j*). Since manager *j*’s expected payoff is positive given \(\sigma\), manager *i* must enter the market as well, contradicting the equilibrium assumption. \(\square\)

### 1.4 Proof of Theorem 2

###
*Proof*

Fix an RS manager *i*. In case \(i=k\), one can fix \(C=0\) to show that all managers are active. Otherwise, consider \(i^* = {\mathrm{argmax}}_{j>i} \alpha _j\) and fix \(\frac{C}{\lambda w} \in (\alpha _{i^*},\alpha _i)\). We start by proving the existence of an equilibrium with active managers \(S_i\).

Consider the strategy profile \(\sigma = (Y_1,\dots ,Y_i)\). For every active manager \(j \leqslant i\), the GS *f* is linearly increasing in \(\sigma _j\), thus \(Y_j\) is still a dominant strategy of manager *j*. If \(i=1\), then the expected payoff of manager *i* is strictly positive, as no deposit is needed. Otherwise, the expected payoff of manager \(1<i<k\) under \(\sigma\) is

where the inequality follows from the choice of *C*. By Lemma 1, we get \(\pi _j > \pi _i\) for every active manager \(j<i\) due to the reduced benchmark and the excess expected return (relative to manager *i*). Thus, no active manager \(j \in S_i\) can gain from becoming inactive.

On the other hand, in case an inactive manager \(j \notin S_i\) becomes active and invests in \(Y_j\),

where the first inequality follows from the reduced benchmark, and the second follows from the choice of *C* and the RS condition. Hence, no manager has a profitable deviation from \(\sigma\), which establishes the existence of an equilibrium.

The last inequality also ensures that the set of active managers is a subset of \(S_i\). Lemma 1 proves that, in every equilibrium with an active manager \(j>i\),^{Footnote 16} all the managers of higher ability than manager *j* must be active as well. If *j* is potentially the highest-index active manager, then his expected payoff would be \(\lambda w v_j - C\tfrac{v_j}{\alpha _j} <0\), and therefore actively investing in \(Y_j\) cannot be an equilibrium strategy. This concludes the first part of the proof.

For the second direction (of the “if and only if” statement), fix \(C < \lambda w \alpha _i\) such that the stated condition holds. Assume, by contradiction, that manager *i* is not RS. Hence, there exists \(j^*>i\) such that \(\alpha _{j^*} \geqslant \alpha _i > \tfrac{C}{\lambda w}\). Fix the profile of optimal portfolios \(\sigma '=(Y_1,\dots ,Y_{j^*})\), and note that

where the inequality follows from \(\alpha _{j^*} > \tfrac{C}{\lambda w}\). By the proof of Lemma 1 and given \(\sigma '\), we know that the expected payoff of every manager superior to \(j^*\) is also positive. Therefore, there exists an equilibrium with managers \(j \notin S_i\), contradicting our preliminary assumption. This concludes the proof of the “if and only if” statement.

We now prove the equivalence between \(\alpha _i > \alpha _{i+1}\) and the existence of an equilibrium \(\sigma =(Y_1,\dots ,Y_i)\). Assume \(\alpha _i > \alpha _{i+1}\) and fix \(\frac{C}{\lambda w} \in (\alpha _{i+1},\alpha _i)\). Relying on the previous part of the proof and given \(\sigma\), we know that no active manager can gain from deviating and becoming inactive. Also, Inequality (3) still holds when taking \(j=i+1\). Thus, we only need to prove that every inactive manager \(j>i+1\) cannot gain from deviating. In case an inactive manager \(j>i+1\) becomes active,

where the first inequality holds because \(v_j< v_{i+1}\) as \(j>i+1\). Therefore, \(\sigma\) is indeed an equilibrium.

Now assume, by contradiction, that \(\sigma\) is an equilibrium and \(\alpha _{i+1}> \alpha _i\). By Inequality (2) we know that \(\alpha _{i+1}\geqslant \alpha _i > \tfrac{C}{\lambda w}\). Hence, if manager \(i+1\) becomes active, then by a similar computation to Inequality (2) we get

This establishes that \(\sigma\) is not an equilibrium and concludes the proof. \(\square\)

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Lagziel, D., Lehrer, E. Transferable deposits as a screening mechanism.
*Econ Theory* **71**, 483–504 (2021). https://doi.org/10.1007/s00199-020-01264-2

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DOI: https://doi.org/10.1007/s00199-020-01264-2