SlowGuess/NeurIPS_2023_OCR huggingface dataset:
A note on ties in SBA. Denote the target action by  $i$ . To simplify presentation, the algorithm presentation in Section 3.4 implicitly assumes that required budgets  $\| t^{*}(i',i)\|_{\infty}$  are distinct for every pair of actions  $(i',i)$ , and therefore the return value of SBA is well-defined. In case of ties, the return value is not well-defined (as the iteration order is not well-defined), but small a modification to the algorithm allows ties to be broken explicitly without affecting other properties of the algorithm: In case of ties in optimal required budget, the soundness of the algorithm and the all-of-nothing property of the returned contracts is not affected. However, the exact functional form of the returned contract may be affected by the iteration order. In case such issue becomes relevant, the SBA algorithm can be modified to first collect a set of optimal contracts (instead of immediately returning when an optimal feasible contract is found), and then select one of the optimal contracts based on the desired criteria. As the proof of Proposition 2 does not depend on iteration order, soundness will not be affected, and worst-case time complexity will remain the same.
B.7 MLRP
In this section, we explore contract design under the MLRP assumption, and prove Theorem 4. We first recall the statistical notion of monotone likelihood ratio:
Definition 11 (Mononote Likelihood Ratio - MLR). The distributions  $P, Q \in \Delta(\Omega)$  have Monotone Likelihood Ratio when  $\frac{Q_j}{P_j}$  is monotonically increasing for all  $j \in \Omega = \{0, \dots, m\}$ . We denote this by  $P \prec Q$ .
An introduction to MLR and its relation to statistical hypothesis testing is provided in Lehmann et al. [40, Section 3.4]. Using this notation, we can reformulate Definition 1:
Definition 12 (MLRP; reformulation of Def. 1 using MLR notation). A principal-agent problem satisfies the Monotone Likelihood Ratio Property if  $F_{i'} \prec F_i$  for every pair of actions  $i, i'$  such that  $c_{i'} < c_i$ .
Claim 10. Let  $P, Q \in \Delta([m])$  such that  $P \prec Q$ . Then  $P_0 > Q_0$  and  $P_m < Q_m$ .
Proof. For the first inequality, assume by contradiction that  $Q_0 \geq P_0$ . Combining with the MLR property, this assumption implies that  $Q_j \geq P_j$  for all  $j \in \{0, \ldots, m\}$ . As the outcome distributions  $P, Q$  are distinct, there exists at least one  $j$  for which  $Q_j > P_j$ . Summing over  $j$  yields:  $\sum_{j \in \Omega} Q_j > \sum_{j \in \Omega} P_j$ , which is a contradiction, as the inequality is strict while both sides sum to 1. The proof for the second inequality follows in the same way.
Definition 13 (MLR crossing point  $j_{P,Q}^{*}$ ). Let  $P, Q \in \Delta(\Omega)$  such that  $P \prec Q$ . The crossing point of  $Q$  over  $P$  is the outcome  $j^{*} \in \{0, \dots, m\}$  such that  $Q_{j^{*}-1} < P_{j^{*}-1}$  and  $Q_{j^{*}} \geq P_{j^{*}}$ . When distribution are not clear from context, we denote  $j^{*} = j_{P,Q}^{*}$
In words, the crossing point  $j^{*}$  is the outcome such that for every lower outcome,  $P$  is strictly more likely than  $Q$ , and starting with this outcome  $Q$  is weakly more likely. By claim 10, the MLR crossing point  $j^{*}$  is uniquely defined for any  $P \prec Q$ , and it holds that  $j^{*} > 0$ .
Definition 14 (Survival function). Given  $P \in \Delta([m])$ , the survival function  $\mathbb{S}_P(\cdot): \Omega \to [0,1]$  is defined as:
$$
\mathbb {S} _ {P} (j) = \mathbb {P} _ {j ^ {\prime} \sim P} [ j ^ {\prime} > j ] = \sum_ {j ^ {\prime} = j + 1} ^ {m} P _ {i}
$$
Informally, the survival function is 1 minus the distribution's CDF. This function is useful in our context, since the total variation distance between two distributions (as appearing in Theorem 5) can be represented as the difference between their survival functions when they satisfy MLR. The intuitive reason for this is that MLR means the distributions are "single-crossing".
Claim 11 (Total variation under MLR). Let  $P, Q \in \Delta(\Omega)$  such that  $P \prec Q$ . It holds:
$$
\| P - Q \| _ {\mathrm {T V}} = \mathbb {S} _ {Q} (j _ {P, Q} ^ {*} - 1) - \mathbb {S} _ {P} (j _ {P, Q} ^ {*} - 1)
$$
Proof. By Claim 4, it holds that:
$$
\left\| P - Q \right\| _ {\mathrm {T V}} = \sum_ {j \in \{0, \dots , m \}} \left(Q _ {j} - P _ {j}\right) ^ {+}
$$
Using the  $j^*$  notation (see Definition 13), we obtain:
$$
\sum_ {j \in \{0, \dots , m \}} \left(Q _ {j} - P _ {j}\right) ^ {+} = \sum_ {j = j _ {P, Q} ^ {*}} ^ {m} Q _ {j} - P _ {j}
$$
And the result is obtained by applying the definition of survival function (see Definition 14).
B.7.1 Two-action contracts with MLRP
Combining Theorem 5 with Claim 10 leads to a characterization of threshold contracts in the case of two actions:
Claim 12. For any two-action contract design problem satisfying MLRP, the optimal contract is a threshold contract:
$$
t _ {j} ^ {*} = \frac {c}{\| F _ {2} - F _ {1} \| _ {\mathrm {T V}}} \mathbb {1} \left[ j \geq j _ {F _ {1}, F _ {2}} ^ {*} \right]
$$
where  $j_{F_1,F_2}^* = \min \left\{j\in \{0,\ldots ,m\} \mid F_{2,j}\geq F_{1,j}\right\}$  as in Definition 13.
Proof. Combining the result of Claim 10 with the monontonicity assumption, the likelihood ratio  $F_{2,j} / F_{1,j}$  crosses 1 exactly once. Denote the crossing point by  $j^*$ , and apply Theorem 5 to obtain the optimal contract.
B.7.2 Two-outcome contracts with MLRP
When there are more than two actions, assume without loss of generality that the contract aims to implement the last action  $n$ . The following claim establishes the existence of theshold contracts in the two-outcome setting  $|\Omega| = 2$ . In contrast to corollary 1, the proof is constructive, and yields a concrete contract:
Claim 13. For any contract design problem satisfying MLRP with  $n > 2$  actions and  $m = 2$  outcomes, the optimal min-budget contract is a threshold contract.
Proof. By Claim 2, the dual LP for two outcomes is given by:
$$
\max_{\lambda \in \mathbb{R}^{n - 1}_{\geq 0},\mu \in \mathbb{R}^{2}_{\geq 0}}\sum_{i = 1}^{n - 1}(c_{n} - c_{i})\lambda_{i}
$$
$$
\forall j \in \{1, 2 \}: \sum_ {i = 1} ^ {n - 1} \left(F _ {n, j} - F _ {i, j}\right) \lambda_ {i} \leq \mu_ {j} \tag {33}
$$
$$
\sum_ {j \in \{1, 2 \}} \mu_ {j} \leq 1
$$
From Claim 10, we obtain that  $F_{n,1} - F_{i,1} < 0$  and  $F_{n,2} - F_{i,2} > 0$  for all  $i \in [n-1]$ , and therefore the first constraint in Eq. (33) is always satisfied for  $j = 1$ . Simplifying Eq. (33) we obtain:
$$
\max_{\lambda \in \mathbb{R}_{\geq 0}^{n - 1},\mu \in \mathbb{R}_{\geq 0}}\sum_{i = 1}^{n - 1}(c_{n} - c_{i})\lambda_{i}
$$
s.t. (34)
$$
\sum_ {i = 1} ^ {n - 1} \left(F _ {n, 2} - F _ {i, 2}\right) \lambda_ {i} \leq 1
$$
which is maximized by allocating all budget to the  $\lambda_{i}$  maximizing the "bang for buck". The dual objective is therefore given by:
$$
\begin{array}{l} B ^ {*} = \max  _ {\lambda \in \mathbb {R} _ {\geq 0} ^ {n - 1}, \mu \in \mathbb {R} _ {\geq 0}} \sum_ {i \in [ n - 1 ]} \left(c _ {n} - c _ {i}\right) \lambda_ {i} \tag {35} \\ = \max _ {i \in [ n - 1 ]} \frac {c _ {n} - c _ {i}}{F _ {n , 2} - F _ {i , 2}} \\ \end{array}
$$
To see that a threshold contract is optimal, let  $t_j^* = B^* \cdot \mathbb{1}[j = 1]$ . Primal objective is  $B^*$ , and the contract is feasible if the primal LP is feasible. The (IC) constraint in Eq. (6) can be written as:
$$
\forall i \in [ n - 1 ]: \frac {\sum_ {j = 1} ^ {2} \left(F _ {n , j} - F _ {i , j}\right) t _ {j}}{c _ {n} - c _ {i}} \geq 1
$$
and indeed for every action  $i\in [n - 1]$
$$
\begin{array}{l} \frac {\sum_ {j = 1} ^ {2} \left(F _ {n , j} - F _ {i , j}\right) t _ {j}}{c _ {n} - c _ {i}} = \frac {F _ {n , 2} - F _ {i , 2}}{c _ {n} - c _ {i}} B ^ {*} \\ = \frac {F _ {n , 2} - F _ {i , 2}}{c _ {n} - c _ {i}} \max  _ {i \in [ n - 1 ]} \frac {c _ {n} - c _ {i}}{F _ {n , 2} - F _ {i , 2}} \\ \geq 1 \\ \end{array}
$$
And therefore  $t_j^*$  is feasible in the primal problem, and also optimal by strong LP duality. Also note that the resulting contract coincides exactly with the optimal min-pay contract as attained by Dütting et al. [19, Lemma 7].
B.7.3 General contracts with MLRP
In this section, we explore min-budget contracts in MLRP settings with more than two actions and more than two outcomes. We start with a negative example, showing that the MLRP assumption is not sufficient for establishing the optimality of threshold contracts:
Claim 14. For  $|\Omega| > 2$ , there exists a design problem satisfying MLRP for which the optimal contract is not a threshold.
Proof. Consider the following contract design problem:
$$
F _ {1} \sim \operatorname {B i n o m i a l} (1 0, 0. 5) \quad c _ {1} = 0
$$
$$
F _ {2} \sim \text {B i n o m i a l} (1 0, 0. 6 5) \quad c _ {2} = 0. 4 5
$$
$$
F _ {3} \sim \operatorname {B i n o m i a l} (1 0, 0. 8) \quad c _ {3} = 1
$$
Figure 7: Graphical illustration of Claim 14. (Left) Outcome distributions  $F_{1}, F_{2}, F_{3}$  and MLR crossing point  $j_{F_2, F_3}^*$ . (Center) The min-budget contract  $t_j^*$ , given by numerically solving Eq. (11) (purple), and numerically solving Eq. (11) while imposing integer constraints  $\phi_j \in \{0, 1\}$  (cyan). Note that the fractional LP solution achieves lower max payout. (Right) Crossing-point survival  $s_i = \mathbb{S}_{F_i}(j_{F_i, F_n}^* - 1)$  as a function of cost  $c_i$  (see Definition 15). Note that the function is not concave, thus the sufficient condition given in Theorem 4 does not hold in this case.
The distributions are binomial, and therefore the contract satisfies MLRP. Numerically solving Eq. (11) yields the following fractional contract:
$$
t _ {\mathrm {L P}} ^ {*} = (0, 0, 0, 0, 0, 0, 0, 1. 1, 1. 4 6, 1. 4 6, 1. 4 6)
$$
Numerically solving Eq. (11) while imposing integer constraints  $\phi_j \in \{0, 1\}$  yields the following contract, which has higher max payout:
$$
t _ {\mathrm {I P}} ^ {*} = (0, 0, 0, 0, 0, 0, 0, 1. 5 1, 1. 5 1, 1. 5 1, 1. 5 1)
$$
Thus for this case any threshold contract is min-budget suboptimal. A graphical illustration of the proof is provided in Fig. 7.
Remark 2. The proof of Claim 14 is provided with 10 outcomes  $(|\Omega| = 10)$  for ease of graphical interpretation, however a minimal counterexample may also be constructed using only 3 outcomes. For example, consider the following design problem:
$$
F _ {1} = (0. 5, 0. 3, 0. 2) \quad c _ {1} = 0
$$
$$
F _ {2} = (0. 3, 0. 4, 0. 3) \quad c _ {2} = 0. 4 5
$$
$$
F _ {3} = (0. 1, 0. 3 5, 0. 5 5) \quad c _ {3} = 1
$$
The proof for this case follows in the same way, where numerical computation yields the contracts:
$$
t _ {\mathrm {L P}} ^ {*} = (0, 1. 9, 2. 6)
$$
$$
t _ {\mathrm {I P}} ^ {*} = (0, 2. 7, 2. 7)
$$
Remark 3. We also note that a counterexample with 2 outcomes is not possible due to Claim 13.
The negative example shows that even with MLRP, the optimal contract incentivizing the highest implementable action needs to rule out deviations of the agent to all other actions, and not just to the second-highest one. This makes the contract complex. We identify a natural economic condition that is sufficient for considering only a single deviation (to the second-highest action), resulting in a simple contract. Note that considering a single such deviation is equivalent to restricting the action space to actions  $\{n - 1, n\}$ .
B.8 Sufficiency condition for more than two actions
For the following definition, recall the definition of MLR crossing point  $j^*$  (Def. 13), and survival function  $\mathbb{S}$  (Def. 14):
Definition 15 (Concave-MLRP; Formal statement of Definition 2). For a contract design setting satisfying MLRP, denote by  $j^{*} = j_{F_{n}, F_{n - 1}}^{*}$  the crossing point of the outcome distribution corresponding to the highest action  $F_{n}$ , and the outcome distribution corresponding to the second-highest action  $F_{n - 1}$ . For any action  $i \in [n]$ , the crossing-point survival  $s_i$  is defined as:
$$
s _ {i} = \mathbb {S} _ {F _ {i}} (j ^ {*} - 1)
$$
In words,  $s_i$  is the probability to get an outcome at or above crossing-point  $j^*$  according to outcome distribution  $F_i$ . From an economic perspective, for every action  $i \in [n]$ ,  $s_i$  is the agent's probability to receive any nonzero payment from choosing action  $i$ , if the contract is designed by restricting the action-space to actions  $\{n - 1, n\}$ . Note however that the definition of  $s_i$  only depends on the outcome distribution  $F_{ij}$ , and does not require solving the contract design problem.
B.8.1 Binomial power-law curves satisfy Concave-MLRP
Recent theoretical work on learning curves has focused mainly on power-law expected learning curves of the form  $a_{n} = 1 - \beta n^{-\gamma}$  [11, 33]. In addition, these curves were also found to provide a good fit for certain practical applications [49, POW2]. In this section, we show that a stochastic generalization of these curves satisfies the Concave-MLRP property:
Definition 16 (Power-law stochastic learning curve). Let  $\beta, \gamma \in \mathbb{R}_{\geq 0}$ , and  $m \in \mathbb{N}$ . A delegated learning setting with actions  $\mathcal{A}$  has a realizable power-law stochastic learning curve with parameters  $\beta, \gamma$  if  $F_{i} = \operatorname{Binomial}\left(1 - \beta n_{i}^{-\gamma}, m\right)$  for all  $n_{i} \in \mathcal{A}$ .
For the proof, we also recall the definition of the regularized incomplete beta function:
Definition 17 (Regularized incomplete beta function). For  $k_{1}, k_{2} \geq 1$ , the regularized incomplete beta function is defined as:
$$
I _ {p} (k _ {1}, k _ {2}) = \frac {\int_ {0} ^ {p} t ^ {k _ {1} - 1} (1 - t) ^ {k _ {2} - 1} \mathrm {d} t}{\int_ {0} ^ {1} t ^ {k _ {1} - 1} (1 - t) ^ {k _ {2} - 1} \mathrm {d} t}
$$
We also recall that  $I_{p}$  is related to the survival function of the binomial distribution [e.g., 1, 6.6.4]:
$$
I _ {p} (k + 1, n - k) = \mathbb {P} _ {x \sim \operatorname {B i n o m i a l} (p, n)} [ x > k ]
$$
Claim 15. Let  $\beta, \gamma \in \mathbb{R}_{\geq 0}$ ,  $m \in \mathbb{N}$ . A delegated learning setting with a power-law stochastic learning curve  $F_{i} = \mathrm{Binomial}(1 - \beta n_{i}^{-\gamma}, m)$ , action costs  $c_{i} = n_{i}$ , and  $\min_{i} n_{i} \geq \left(\frac{\beta(m + \gamma^{-1})}{1 + \gamma^{-1}}\right)^{\frac{1}{\gamma}}$  satisfies the Concave-MLRP assumption.
Proof. Denote the expected accuracy of each action by  $a_{n} = 1 - \beta n^{-\gamma}$ . The survival function of the binomial distribution is given by:
$$
s _ {n} = I _ {a _ {n}} \left(j ^ {*}, m + 1 - j ^ {*}\right)
$$
Where  $I_{a_n}$  is the regularized incomplete beta function (Definition 17). With slight abuse of notation, we treat  $a_n$  and  $s_n$  as continuous functions of  $n$ . Taking the derivative by  $n$  and ignoring the constant denominator, we obtain:
$$
\frac {\mathrm {d} s _ {n}}{\mathrm {d} n} \propto a _ {n} ^ {j ^ {*} - 1} (1 - a _ {n}) ^ {m - j ^ {*}} \frac {\mathrm {d} a _ {n}}{\mathrm {d} n}
$$
Plugging the functional form of  $a_{n}$ :
$$
\begin{array}{l} \frac {\mathrm {d} s _ {n}}{\mathrm {d} n} \propto \left(1 - \beta n ^ {- \gamma}\right) ^ {j ^ {*} - 1} \left(\beta n ^ {- \gamma}\right) ^ {m - j ^ {*}} \beta \gamma n ^ {- (\gamma + 1)} \\ \propto \left(\beta n ^ {- \gamma}\right) ^ {m - j ^ {*} + 1 + \gamma^ {- 1}} \left(1 - \beta n ^ {- \gamma}\right) ^ {j ^ {*} - 1} \\ \end{array}
$$
As a function of  $\beta n^{-\gamma}$ , the function  $\frac{\mathrm{d}s_n}{\mathrm{d}n}$  attains its extremum at:
$$
\beta \tilde {n} ^ {- \gamma} = \frac {m + 1 - j ^ {*} + \gamma^ {- 1}}{m + \gamma^ {- 1}}
$$
$$
\tilde {n} = \left(\frac {\beta (m + \gamma^ {- 1})}{m + 1 - j ^ {*} + \gamma^ {- 1}}\right) ^ {\frac {1}{\gamma}}
$$
And therefore the function  $s_n$  has an inflection point at  $\tilde{n}$  (see Fig. 8). From the upper bound  $j^* \leq m$  we obtain:
$$
\tilde {n} \leq \left(\frac {\beta (m + \gamma^ {- 1})}{1 + \gamma^ {- 1}}\right) ^ {\frac {1}{\gamma}} = n _ {0}
$$
And as  $\min_i n_i \geq n_0$ , the function  $s_n$  is convex as a function of  $c_n$  for all  $n \in \mathcal{A}$ .
The proof is illustrated in Fig. 8.
Figure 8: Graphical intuition for the sufficiency condition in Claim 15. (Left) Expected accuracy curve for MNIST-784 MLP. Blue dots represent empirical data from the LCDB dataset, cyan curve represents power-law fit  $(a_{n} = 1 - \beta n^{-\gamma}$ , with  $\hat{\beta} = 1.89$ ,  $\hat{\gamma} = 0.33)$ . (Center) Crossing point survival  $s_i$  as a function of cost  $c_{i}$  (see Definition 15), for  $m = 30$ . Cyan dot represents inflection point  $\tilde{n} \approx 2022$ . Red vertical line represents the inflection point bound  $n_0 \approx 3957$  suggested by the proof. It can be observed that the curve is concave for all  $n > \tilde{n}$ . (Right) Second derivative of the survival function  $s_n$ , illustrating the position of the bound  $n_0$  in relation to the inflection point  $\tilde{n}$ .
B.8.2 Proof of sufficiency theorem
Theorem 6 (Concave-MLRP implies threshold contracts; formal statement of Theorem 4). For a contract design problem satisfying MLRP, consider  $s_i$  as a function of cost  $c_i$ . If this function is concave, then the optimal contract is a threshold contract. Furthermore, the contract is successfully recovered by the SBA algorithm.
Proof. To prove this claim, we construct a relaxed version of the min-budget LP (Eq. (6)), and apply Theorem 5 in order to solve it. We then show that this solution is also feasible for the original LP.
Given the min-budget LP, construct a relaxed LP by eliminating (IC) constraints for all  $i \leq n - 2$ :
$$
\forall j \in \Omega : t _ {j} \leq B \tag {36}
$$
$$
\sum_ {j \in \Omega} F _ {n - 1, j} t _ {j} - c _ {n - 1} \leq \sum_ {j \in \Omega} F _ {n, j} t _ {j} - c _ {n}
$$
Eq. (36) only depends on  $F_{n}$  and  $F_{n - 1}$ , and therefore claim 12 can be applied to obtain the optimal min-budget contract:
$$
t _ {j} ^ {*} = \frac {c _ {n} - c _ {n - 1}}{\left\| F _ {n} - F _ {n - 1} \right\| _ {\mathrm {T V}}} \mathbb {1} \left[ F _ {n, j} \geq F _ {n - 1, j} \right] \tag {37}
$$
Let  $j^{*} = \min \{j\in \{0,\ldots ,m\} \mid F_{n,j}\geq F_{n - 1,j}\}$  as in Definition 13, and denote  $s_i = \mathbb{S}_{F_i}(j^* -1)$ . By claim 11, we obtain:
$$
\left\| F _ {n} - F _ {n - 1} \right\| _ {\mathrm {T V}} = s _ {n} - s _ {n - 1}
$$
For all  $i <   n$  , denote:
$$
b _ {i} = \frac {c _ {n} - c _ {i}}{s _ {n} - s _ {i}}
$$
Using the definition of  $b_{i}$ , Eq. (37) can be rewritten as:
$$
t _ {j} ^ {*} = b _ {n - 1} \mathbb {1} [ j \geq j ^ {*} ]
$$
By assumption,  $s_i$  is a concave function of  $c_i$ , and therefore the function  $b_i$  is monotonically non-decreasing for all  $i < n$ :
$$
b _ {n - 1} \geq b _ {i}
$$
Dividing both sides by  $b_{i}$ , we obtain:
$$
\frac {b _ {n - 1}}{b _ {i}} \geq 1
$$
Figure 9: Graphical intuition for the sufficiency condition in Theorem 4 (restated in Theorem 6). (Left) Outcome distribution in a contract design setting satisfying MLRP, generated by a series of binomial distributions with power-law expectation  $\mathrm{acc}_D(h_n) = 0.9 - 0.4\left(\frac{n}{100}\right)^{-0.3}$ . The gray dotted line represents the MLR crossing point  $j_{F_4,F_5}^*$  (see Definition 13). (Center) Crossing point survival  $s_i$  as a function of cost  $c_i$  (see Definition 15). It can be observed that the curve is concave. (Right) The min-budget contract implementing action 5. It is a threshold contract, as guaranteed by Theorem 4. Compare to Fig. 7, where the sufficiency condition does not hold.
Plugging in the definition of  $b_{i}$ , and multiplying both sides by  $(c_{n} - c_{i})$ :
$$
b _ {n - 1} \left(s _ {n} - s _ {i}\right) \geq c _ {n} - c _ {i} \tag {38}
$$
By definition of  $t_j^*$ , the expected payout of contract  $t_j^*$  under action  $i' \in [n]$  is given by:
$$
\sum_ {j = 0} ^ {m} t _ {j} ^ {*} F _ {i ^ {\prime}, j} = b _ {n - 1} \sum_ {j = j ^ {*}} ^ {m} F _ {i ^ {\prime}, j} = b _ {n - 1} s _ {i ^ {\prime}} \tag {39}
$$
Plugging Eq. (38) into Eq. (39) yields:
$$
\sum_ {j = 0} ^ {m} t _ {j} F _ {n, j} - \sum_ {j = 0} ^ {m} t _ {j} F _ {i, j} \geq c _ {n} - c _ {i} \tag {40}
$$
As Eq. (40) is identical to the (IC) constraint in Eq. (6), we obtain that  $t_j^*$  satisfies all the (IC) constraints in the original LP. From this we conclude that  $t_j^*$ , which is a threshold contract, is also an optimal contract with the respect to the full min-budget LP.
For the second part of the claim, note that the SBA algorithm also attempts to solve Eq. (36) on the iteration corresponding to action  $N - 1$ . As the solution of Eq. (36) is guaranteed to be feasible for the original LP under Concave-MLRP, the SBA successfully recovers it.
B.9 Min-budget contracts with guaranteed minimum payout
Our problem setting assumes that the agent selects its action rationally, as described in Eq. (2). However, in some practical settings the agent may be risk averse to some extent, and require guaranteed minimum payment from the contract. In this section, we show that the rationality assumption is made without loss of generality in such cases: Lemma 5, which we prove below, shows that any optimal contract with guaranteed minimum payout can be represented as the sum of a min-budget contract without guaranteed payout, and a constant representing the payout guarantee.
Definition 18 (Guaranteed minimum payout). Let  $\delta \geq 0$ . A contract  $t \in \mathbb{R}_{\geq 0}^{|\Omega|}$  has guaranteed payout of size  $\delta$  if  $t_j \geq \delta$  for all  $j \in \Omega$ .
Lemma 5. A contract  $t$  is a min-budget contract with guaranteed payout  $\delta$  requiring budget  $B$  if any only if there exist a min-budget contract  $t'$  (without guaranteed payout) requiring budget  $B'$  such that  $t = t' + \delta$  and  $B = B' + \delta$ .
Proof. When agents require a guaranteed minimum payout  $\delta \geq 0$ , we add a constraint to the MIN-BUDGET linear program defined in Eq. (6), such that an optimal min-budget contract  $t$  with
guaranteed payout  $\delta$  is an optimal solution to the following linear program:
$$
\min_{\substack{t\in \mathbb{R}_{\geq 0}^{|\Omega |},B\in \mathbb{R}_{\geq 0}}}B
$$
$$
\forall j \in \Omega : t _ {j} \leq B \tag {41}
$$
$$
\forall j \in \Omega : t _ {j} \geq \delta \quad (\text {M I N W A G E})
$$
To prove the first direction of the equivalence, assume that  $t$  is a min-budget contract with guaranteed payout  $\delta$  and budget  $B$ , and thus an optimal solution of Eq. (41). Define the variable transformation:
$$
t ^ {\prime} = t - \delta \tag {42}
$$
$$
B ^ {\prime} = B - t _ {0}
$$
By Eq. (42), the (BUDGET) constraint in Eq. (41) transforms into:
$$
\forall j \in \Omega : t _ {j} \leq B
$$
$$
\Leftrightarrow t _ {j} ^ {\prime} + \delta \leq B ^ {\prime} + \delta \tag {43}
$$
$$
\Leftrightarrow t _ {j} ^ {\prime} \leq B ^ {\prime}
$$
The (IC) constraint transforms into:
$$
\begin{array}{l} \forall i ^ {\prime} \neq i: \sum_ {j \in \Omega} F _ {i ^ {\prime}, j} t _ {j} - c _ {i ^ {\prime}} \leq \sum_ {j \in \Omega} F _ {i, j} t _ {j} - c _ {i} \\ \Leftrightarrow \sum_ {j \in \Omega} F _ {i ^ {\prime}, j} \left(t _ {j} ^ {\prime} + \delta\right) - c _ {i ^ {\prime}} \leq \sum_ {j \in \Omega} F _ {i, j} \left(t _ {j} ^ {\prime} + \delta\right) - c _ {i} \\ \Leftrightarrow \sum_ {j \in \Omega} F _ {i ^ {\prime}, j} t _ {j} ^ {\prime} - c _ {i ^ {\prime}} + \delta \underbrace {\sum_ {j \in \Omega} F _ {i ^ {\prime} , j}} _ {= 1} \leq \sum_ {j \in \Omega} F _ {i, j} t _ {j} ^ {\prime} - c _ {i} + \delta \underbrace {\sum_ {j \in \Omega} F _ {i , j}} _ {= 1} \tag {44} \\ \Leftrightarrow \sum_ {j \in \Omega} F _ {i ^ {\prime}, j} t _ {j} ^ {\prime} - c _ {i ^ {\prime}} \leq \sum_ {j \in \Omega} F _ {i, j} t _ {j} ^ {\prime} - c _ {i} \\ \end{array}
$$
And the (MINWAGE) constraint transforms into:
$$
\begin{array}{l} \forall j \in \Omega : t _ {j} \geq \delta \tag {45} \\ \Leftrightarrow t _ {j} ^ {\prime} \geq 0 \\ \end{array}
$$
Plugging back the transformed constraints (44, 43, 45) into Eq. (41), we obtain that  $(t', B')$  is an optimal solution of the original MIN-BUDGET linear program Eq. (6), and therefore  $t'$  is an optimal min-budget contract without minimum guaranteed payout.