## 1 Introduction

Wagering mechanisms [3, 4, 5, 6, 7] are one-shot betting mechanisms that allow a principal to elicit participating agents’ beliefs about an event of interest without paying out of pocket or incurring a risk. Compared with prediction-market-type of dynamic elicitation mechanisms, one-shot wagering maybe preferred due to its simplicity. It is particularly designed for agents with immutable beliefs who “agree to disagree” and who do not update their beliefs. In a wagering mechanism, each agent submits a prediction for the event and specifies a wager, which is the maximum amount of money that the agent is willing to lose. Then after the event outcome is revealed, the total wagered money will be redistributed among the participants. Researchers have developed wagering mechanisms with various theoretical properties. In particular, [3, 4] proposed a class of weighted score wagering mechanisms (WSWM) that satisfy a set of desirable properties, including budget balance, individual rationality, incentive compatibility, sybilproofness, among some others^{2}^{2}2Precise definitions can be found in Section 4.. [5] later proposed a no-arbitrage wagering mechanism (NAWM) that removes the opportunities for participating agents to risklessly profit.

However, in both WSWM and NAWM, it has been observed that a participant only loses a very small fraction of his total wager even in the worst case. This seems to be undesirable in practice as it is against the “spirit” of betting and a wager effectively loses its meaning as a budget. [6] first formalized this observation by indicating that these mechanisms are not Pareto optimal, where Pareto optimality requires that there is no profitable side bet among participants before the allocation of a wagering mechanism. Unfortunately, Pareto optimality is not compatible with individual rationality, weak budget balance and weak incentive compatibility, as shown by [6]. A double clinching auction (DCA) wagering mechanism [6] is proposed to improve Pareto efficiency, and parimutuel consensus mechanism (PCM) is shown to satisfy Pareto optimal [7], but violating incentive compatibility.

This paper is another quest of wagering mechanisms with better theoretical properties. We expand the design space of wagering mechanisms to allow randomization on agent payoffs and ask whether we can achieve all aforementioned desirable properties, including Pareto optimality. We give a positive answer to this question: Our randomized wagering mechanisms are the first ones to achieve Pareto optimality along with other properties.

We first show that a simple randomized lottery type of implementation of existing mechanisms (Lottery Wagering Mechanisms (LWM)) satisfy all desirable properties. The idea of LWM is, instead of receiving re-allocated money from a deterministic wagering mechanism, each agent will receive a number of lottery tickets proportional to their payoff from the deterministic wagering mechanism. Then each of the agents will have a chance of winning the entire amount of wagered money (collected from all participants) with a chance being proportional to his number of lottery tickets.

We then study another family of randomized mechanisms which we name as Surrogate Wagering Mechanisms (SWM), that bring insights from learning with noisy data [1, 8] to wagering mechanism design. The idea is to first generate a “surrogate outcome" for each agent according to the true event outcome. His reported prediction will then be evaluated using this surrogate but biased outcome; a bias removal procedure is applied to this score such that in expectation the agent will receive a score that is as if evaluated against the ground truth. That is, though being randomized, SWM preserve the incentive properties of a deterministic wagering mechanism. We show that certainSWM satisfy all desired properties of a wagering mechanism. Notably, SWM are robust to the case when there is only access to a noisy copy of the ground truth - this property is due to the fact that we borrow the machineries from the literature of learning with noisy data. We believe this is another unique contribution to the literature of wagering mechanism design. Most of the presentations of SWM focus on binary outcome events but we do show how to extend the results to multi-outcome () settings.

The rest of this paper is organized as follows. We discuss relevant literature in the Section 2. Section 3 introduces some preliminaries. We define randomized wagering mechanisms as well as desirable theoretical properties for them in Section 4. Section 5 presents a family of lottery-based wagering mechanisms. A family of surrogate wagering mechanisms are introduced in Section 6. We extend surrogate wagering mechanisms on NAWM and on the multi-outcome event settings in Section 7. Extensive simulations are presented in Section 8 to demonstrate the advantages of randomized wagering mechanisms. Section 9 concludes this paper.

## 2 Related works

The ability to elicit *information*, in particular predictions and forecasts about future events, is crucial for many application settings and has been studied extensively in the literature. Proper scoring rules have been designed [9, 10, 11, 12, 13] for this purpose, where each agent is rewarded by how well their reported forecasts predicted the true realized outcome (after the outcome is resolved). Later, competitive scoring rule [14]

and a parimutuel Kelly probability scoring rule

[15] adapt proper scoring rules to group competitive betting. Both mechanisms are budget balanced so that the principal doesn’t need to pay any participant. These spur the further development of the previously discussed wagering mechanisms [3, 4, 5, 6, 7] and the examination of their theoretical properties.Our method used in lottery wagering mechanisms to transfer an arbitrary deterministic wagering mechanism into a randomized one, while maintaining the properties, is inspired by the method proposed in [16]

. The paper studies the incentive compatible forecasting competition and it transfers scores of multiple predictions into the odds of winning to maintain properties of the scoring rules.

[3] proposed a randomization method based on WSWM via randomly selecting strictly proper scoring rules and proper scoring rules with extreme values to increase the stake. However, this method does not generalize to other deterministic wagering mechanisms. [17] proposed to apply differential privacy technology to randomize the payoff of wagering mechanisms in order to preserve the privacy of each agent’s belief. However, their method does not maintain budget balance (in ex-post).The idea of using randomization in wagering mechanism design is not entirely new, but not thoroughly studied. Both [3, 17] proposed certain types of randomized wagering mechanisms, but neither of the mechanisms satisfies Pareto optimality. The randomized wagering mechanisms first appeared in [3]. There, the randomization is restricted to randomly selecting different scoring rules used in WSWM. It introduced this randomization in order to alleviate the the problem that in WSWM, agents only lose a small fraction of their wagers regardless of the event outcome. However, even with this randomization, an agent won’t lose all his wager in the worst when the number of agents is finite. [17] applied differential privacy technology to randomize the payoff of wagering mechanisms. Its goal is to preserve the privacy of agents’ beliefs.

## 3 Preliminaries

In this section, we explain the scenario where a wagering mechanism applies and formally introduce the deterministic wagering mechanisms. Consider a scenario where a principal is interested in eliciting subjective beliefs from a set of agents

about a random variable (event)

, which takes a value (outcome) in set . The belief of each agentis private, denoted as a vector of occurrence probabilities of each outcome

. Following the previous work on wagering mechanism, this paper continues to adopt an immutable belief model for agents. Unlike in a Bayesian model, agents with immutable beliefs do not update their beliefs. The immutable belief model and the Bayesian model are two extremes of agent modeling for information elicitation, with the reality lies in between and arguably closer to the immutable belief side as people do “agree to disagree.” Moreover, [4] showed that while WSWM was designed for agents with immutable beliefs, it continued to perform well for Bayesian agents who have some innate utility for trading.The principal uses a *wagering mechanism* to elicit the private beliefs of agents. In a wagering mechanism, each agent reports a probability vector , capturing his belief, and wager . Similar to [3], we assume that wagers are exogenously determined for each agent and are not a strategic consideration. We use and to denote the reports and the wagers of all agents respectively, and use and to denote the reports and wagers of all agents other than agent .
Besides, we use to denote for any set of agents .
After an event outcome is realized, the wagering mechanism redistributes all the wagers collected from agents according to . The net-payoff of agent is defined as the payoff or the money that agent receives from the redistribution minus his wager. The wagering mechanism defines a net-payoff function for each agent with wager constraint and constraint whenever .
The two constraints ensure that no agent can lose more than his wager and no agent with zero wager can gain.

### 3.1 Strictly proper scoring rules and weighted score wagering mechanisms

*Strictly proper scoring rules* [13] are scoring functions proposed and developed to truthfully elicit beliefs from risk-neutral agents. They are building blocks of many incentive compatible wagering mechanisms, such as WSWM and NAWM.
A strictly proper scoring rule solely rewards a prediction by a score , according to the realization of the random variable . The scoring function is designed such that the expected payoff of truthful reporting is strictly larger than that of any other report, i.e,

There is a rich family of strictly proper scoring functions, including Brier scores (for binary outcome event, , where is agent ’s report of ), logarithmic and spherical scoring functions. Strictly proper scoring rules are closed under positive affine transformations.

*WSWM* [3] rewards an agent according to his wager and the accuracy of his prediction relative to that of other agents’ predictions. The net-payoff of agent in WSWM, is formally defined as

(1) |

where is any strictly proper scoring rule bounded within . Notice that WSWM strictly encourages truthful report of predictions, because the net-payoff of agent is a strictly proper scoring rule of his prediction. Besides, is always zero by the form of the net-payoff formula, no matter what is. That is the budget balance property of Eqn. (1) doesn’t depend on the form of the scoring function. The budget balance result of our randomized wagering mechanisms is also due to this property.

## 4 Randomized wagering mechanisms

We introduce randomized wagering mechanisms as extensions of deterministic wagering mechanisms. Similar to deterministic wagering mechanisms, the net-payoff of an agent depends on all agents’ predictions and wagers , as well as the realized outcome . But different from deterministic wagering mechanisms, the net-payoffs are now random variables. For notational simplicity, we now use to represent the random variable of agent ’s net-payoff in a randomized wagering mechanism. We use to represent the realization of . We use and as abbreviations when are clear in the context. We denote the maximum/minimum possible value of a random variable by /

. We denote the joint distribution of

by and the marginal distribution of by .###### Definition 1.

Given the reports and wagers of agents and the event outcome , a *randomized wagering mechanism*
defines a joint distribution , and pays each agent by a net-payoff , where are jointly drawn from . Meanwhile, and whenever .

A deterministic wagering mechanism is a special case of randomized wagering mechanisms when is a point distribution for all agent .

### 4.1 Desirable properties

In the literature, several desirable properties of wagering mechanisms have been proposed in the deterministic context. [3] introduced (a) individual rationality, (b) incentive compatibility, (c) budget balance, (d) sybilproofness, (e) anonymity, (f) neutrality. [5] introduced (g) no arbitrage property. [6] introduced (h) Pareto optimality. We extend these properties to the randomized context. These new properties reduce to the properties defined in the literature for the special case of deterministic wagering mechanisms.

(a) Individual rationality requires that each agent has nothing to lose in expectation by participating.

###### Definition 2.

A randomized wagering mechanism is individually rational (IR) if , and , there exists such that

(b) Incentive compatibility requires that an agent’s expected net-payoff is maximized when he reports honestly, regardless of other agents’ reports and wagers.

###### Definition 3.

A randomized wagering mechanism is weakly incentive compatible (WIC) if

A randomized wagering mechanism is strictly incentive compatible (SIC) if the inequality is strict.

(c) Ex-post budget balance requires the principal not losing money after the net-payoffs are realized.

###### Definition 4.

A randomized wagering mechanism is weakly ex-post budget-balanced (WEBB) if for any realization of drawn from the joint distribution . A randomized wagering mechanism is ex-post budget-balanced (EBB) if the equality always holds.

(d) Sybilproofness requires that no agent can increase its expected net-payoff by creating fake identities and splitting his wager, regardless of other agents’ reports and wagers.

###### Definition 5.

A randomized wagering mechanism is sybilproof if , , and let and we have

(e) Anonymity requires that agents’ identities do not affect their net-payoffs. Let be a permutation of the set of agents , and denote the reports and wagers of agents after applying the permutation respectively. Denote the joint distribution of net-payoffs of agents in after applying the permutation on agents.

###### Definition 6.

A randomized wagering mechanism is anonymous if

(f) Neutrality requires that the net-payoffs do not depend on the labeling of the event outcomes. Let be a permutation of the set of outcomes . Denote by the reported prediction of agent after we relabel the outcomes according to permutation , and denote by the new label of an outcome .

###### Definition 7.

A randomized wagering mechanism is neutral if

(g) No arbitrage requires that no agent can risklessly make profits.

###### Definition 8.

A randomized wagering mechanism has no arbitrage if such that

(h) Pareto optimality in economics refers to an efficient situation where no trade can be made to improve an agent’s payoff without harming any other’s payoff. In an IR wagering mechanism, agents with different beliefs can always form a profitable wagering game for each of them if they all have a positive budget. [6] defined Pareto optimality of a wagering mechanism as a property that agents with different beliefs will always lose all of their own wagers under at least one of the event outcomes (this “worst case" outcome might be different for different agents). Thus, no agent can commit to secure part of his wager from the mechanism before the event outcome is realized. Thus, before the event outcome is realized, no additional profitable wagering game can be made. We define Pareto optimality for randomized wagering mechanisms in a similar spirit: no agents with different beliefs can commit to secure part of their wagers before the event outcome is realized.

###### Definition 9.

A randomized wagering mechanism is Pareto optimal (PO) if

#### Properties of existing wagering mechanisms

We summarize the properties of existing wagering mechanisms^{3}^{3}3WSWM, NAWM, DCA, PCM, randomized WSWM [3], private WSWM [17] and ours in Table 1. To emphasize, no existing mechanism satisfies all properties (a)-(h). Moreover, [6] showed an impossibility result for deterministic wagering mechanisms that it is impossible to achieve properties IR, WIC, WEBB, and PO simultaneously. For existing randomized wagering mechanisms, the randomized WSWM in [3] only satisfies PO in the limit of large population of participants, and the private WSWM [17] does not satisfy WEBB and PO.

Budget | Incentive | Pareto | No | |

Mechanism | Balance | Compatibility | Optimality | Arbitrage |

WSWM [3] | Strictly | Strictly | False | False |

NAWM [5] | Weakly | Strictly | False | True |

DCA [6] | Strictly | Weakly | False | True |

PCM [7] | Strictly | False | True | True |

Randomized WSWM [3] | Strictly | True | False | True |

Private WSWM [17] | False | True | False | True |

LWS (this paper) | Strictly | True | True | True |

RP-SWME (this paper) | Strictly | True | True | True |

(All of the mechanisms in this table satisfy individual rationality, anonymity, neutrality and sybilproofness.) |

## 5 Lottery wagering mechanisms

In this section we introduce a class of randomized wagering mechanisms, namely *lottery wagering mechanisms*, which extend arbitrary deterministic wagering mechanisms into randomized wagering mechanisms. We are going to show that this set of mechanisms can easily preserve the randomized version of the properties of the underlying deterministic wagering mechanisms, while maintaining Pareto optimality, overcoming the impossibility result.

In lottery wagering mechanisms, each agent receives a number of lottery tickets in proportion to the *payoff* he gets under a deterministic wagering mechanism, and a winner is drawn from all the lottery tickets to win the whole pool of wagers. The mechanisms are designed in a way such that the expected payoff of each agent is the same as that in the deterministic wagering mechanism and each agent has a positive probability to lose all his wager. Hence, no profitable side bet exists and the mechanisms are Pareto optimal.
We formally present the lottery wagering mechanism that extends an arbitrary deterministic wagering mechanism DET in Mechanism 1. To distinguish the payoff from the net-payoff, we denote the payoff by of agent .

Lottery wagering mechanisms are powerful in obtaining desirable theoretical properties. We show in Theorem 1 that the lottery wagering mechanism that extends the WSWM, denoted as LWS, satisfies all properties (a)-(h).

###### Theorem 1.

LWS satisfy all properties (a) - (h).

However, we notice that an agent either loses all his wager or wins the entire pool of wagers, resulting in a high variance in each agent’s payoff. Besides, an agent has a large probability of losing money even if his prediction is much more accurate than the others’. Consider the following example: when agents have uniform wagers, in the

WSWM mechanism, no agent can have a net-payoff that is more than his wager. Consequently, in the LWS mechanism, each agent has at least chance of losing all his wager no matter what predictions other agents report. Ideally, we prefer a randomized wagering mechanism with moderate payoff variance and high probability of winning money for accurate agents. To alleviate these two shortcomings of the LWM, we can always mix a deterministic wagering mechanism with LWM by assigning each of them a probability to be executed. This probabilistic mixture provides us the flexibility of adjusting the variance of payoff and the probability of winning money.## 6 Surrogate wagering mechanisms

In this section, we propose the *surrogate wagering mechanisms (SWM)*. We first introduce the general SWM, then variants of SWM which obtain desirable properties and have moderate variance in payoffs and a large probability of winning for accurate predictions. We then notice that randomization opens up the possibility of dealing with situations where only noisy ground truth is available. We discuss how to extend our results to this noisy setting.

### 6.1 Generic surrogate wagering mechanisms

A surrogate wagering mechanism consists of three main steps: (1) the SWM generates a surrogate event outcome for each agent based on the true event outcome and a randomization device; (2) the SWM

evaluates each agent’s prediction according to the surrogate event outcome using a designed scoring function such that the score is an unbiased estimate of the score derived by applying a strictly proper scoring rule to the ground truth outcome; (3) the

SWM applies WSWM to the scores based on the surrogate event outcome to determine the final net-payoff of each agent. Next, we explain the three steps in details. For clarity and simplicity of exposition, we consider only binary events, i.e., , in this section^{4}

^{4}4Extension to multi-outcome events can be found in our supplemental materials.. We will use to denote agent ’s report for , which fully describes for binary events.

#### Step 1. Surrogate event outcomes

A SWM generates a surrogate event outcome for each agent . Denote . s are drawn independently conditional on , and are specified by the SWM. The conditional marginal distribution can be expressed by two parameters, the error rates of the surrogate outcome: and . The conditional marginal distribution can be any distribution satisfying . We use and to denote the realization of and respectively.

#### Step 2. Computing unbiased scores

Given a strictly proper scoring rule within [0,1], the SWM computes the score of an agent as , where

(2) |

is the realized surrogate event outcome for agent . Lemma 1 shows that is an unbiased operator on the score in the sense that

###### Lemma 1 (Lemma 3.4 of [2]).

, we have

###### Corollary 1.

Given an event , a prediction , a strictly proper scoring rule , error rates where and a surrogate event randomly generated in a way that and , the scoring rule given by is a strictly proper scoring rule in expectation in the sense that .

#### Step 3. Computing net-payoffs

In the final step, the SWM computes the net-payoff of agent using WSWM and the unbiased score of agent , i.e., replacing score in Eqn. 1 by score . Formally, we have

(3) |

and and are the event outcome and the surrogate event outcome for each agent respectively.

We formally present SWM in Mechanism 2. According to our Corollary 1 (applying to each score terms), we have
Because the deterministic WSWM satisfies properties ((a)-(f)) [3],
SWM also satisfies these properties. Meanwhile, a realization of the score can be larger than 1, offering a chance for agent to lose (or win) more than the money he can lose (or win) in deterministic WSWM. However, we also notice that for some extreme values of error rates, the constraint can be violated^{5}^{5}5For example, in a wagering game, two agents both wager 1 and report and , respectively. Let . In the worst case of agent 1, the surrogate outcomes are realized as . Then, ., i.e., an agent may lose more than he wager, which makes SWM invalid. In the next section, we show that by selecting error rates in a subtle way, we can obtain all the properties (a)-(h) without violating the wager constraint .

### 6.2 Swm with Error rate selection (Swme) and random partition Swme (Rp-Swme)

We notice that according to Corollary 1, no matter which error rates , are chosen, the unbiasedness property of SWM holds, i.e., . In other words, we can choose the error rates in an arbitrary way (even in an ex-post way) without changing the expected net-payoff^{6}^{6}6The expectation is taken over the randomness of the mechanism conditioned on the event outcome. of each agent under any realized event outcome.
This gives us the flexibility to tune the maximal amount of money each agent can win or lose in the game, while preserving the properties ((a)-(f)) inherited from WSWM.

Given reports and wagers but not the event outcome , the error rate pair that guarantees no wager violation under any outcome and any realization of the randomness induced by SWM may not be unique. We propose Algorithm 3 to select a pair of error rates after the reports and wagers are collected, such that at least one agent loses all his wager in the worst case of the outcome and the realization of randomness in SWM. We name the mechanisms SWME when we use Algorithm 3 to select the error rates for SWM.

###### Lemma 2.

SWME has no wager violation and when there exists at least one report , at least one of the agents loses all his wager in the worst case w.r.t. the event outcome and the randomness of SWME.

### 6.3 Proof of Lemma 2

###### Proof.

In this proof, we use Brier Score as the scoring rule used by the mechanism, i.e., , and is agent ’s report of . The proof can be extended to other strictly proper scoring rule within [0, 1].

We first consider the corner case where all agents reports 0.5. It can be verified that in Algorithm 2, , and the algorithm sets and SWME is reduced to WSWM. Thus, no wager violation happens.

Next, we consider the scenario that . In this scenario, we first prove that, in Algorithm 2 .

We have (the equality only holds when ), . Let

and

We have , and (there exists at least one agent that ). Therefore, . We have .

Next, we prove that if let be a variable, and let , the worst cast net-payoff (w.r.t. the event outcome and the randomness of the mechanism) of agent is a decreasing function of .

In the worst case of agent , and . We have . Therefore, is decreasing with .

Finally, it is easy to verify that when , .

Therefore, when we set for each agent , , no agent can lose more than his wager and agent loses all his wager in the worst case. ∎

Note Lemma 2 doesn’t imply PO for SWME - if there exist two agents who have different predictions and have wager left even in their own worst cases, they can form a profitable bet against each other. We propose a variant of SWME to fix this caveat in next subsection.

#### Random partition Swme (Rp-Swme)

Lemma 2 implies that when agents are partitioned into groups of two, there will not exist side bets. Also a smaller number of agents imposes less restrictions in selecting the error rates, and thus each agent’s wager can be fully leveraged in the randomization step. We would like to note that this is a very unique property of SWME: as both shown in [6] and our experimental results, existing wagering mechanisms (including DCA) incurs low risk when the number of agents is small. This not only implies that SWME is particularly suitable for small group wagering but also points out a way of further improving the risk property of SWME, i.e. via randomly partitioning agents into smaller groups. We formally present the random partition SWME in Mechanism 4. We will show in next Section that the random partitioning achieves all properties (a)-(h).

### 6.4 Properties of Swme and Rp-Swme

###### Theorem 2.

Both (SWME) and (RP-SWME) satisfy properties (a)-(g). (RP-SWME) satisfies (h).

###### Proof.

We provide full proofs in Section 2.4 of the supplemental material, but we give the arguments for establishing surrogate wager’s ex-post budget balance (despite of the randomness), incentive compatibility, and Pareto optimality.

#### (a) Individual rationality and (b) (strictly) incentive compatibility

First consider SWME. For an arbitrary profile of reports and wagers , Algorithm 3 outputs an profile of error rates of all agents. Denote by the corresponding surrogate function specified using the error rate profile for agent . For each and :

using Corollary 1. Then (using linearity of expectation, note encode the randomness in )

Note the above holds for any possible reports (). Thus the incentive properties of WSWM will preserve, i.e., IR and SIC, The proof for RP follows immediately, as RP-SWME first runs a random partition, which does not depend on agents’ reports and wagers, and then runs SWME for each group of agents after the partition.

#### (c) Ex-post budget balance

This can be shown via writing down the sum of net-payoffs defined in Eqn. (3). Our note below Eqn. (1) also states that the budget balance property doesn’t depend on the specific forms of the scoring functions therein. We formally present the deduction as follows:

The above also shows that for each group from the random partition of (RP-SWME), ex-post budget balance is satisfied. Thus, we also proved ex-post budget balance for (RP-SWME).

#### (d) Sybilproofness:

SWME is sybilproof as we proved that the expected payoffs of SWME are the same as WSWM (and WSWM is Sybilproof). RP-SWME is also Sybilproof as a corollary of Lemma 3.

###### Lemma 3.

If a (randomized) wagering mechanism is (weakly) budget-balanced, (weakly) incentive compatible, Sybilproof, then the mechanism that first uniformly randomly pairs agents in groups of two and then runs mechanism for each group is still Sybilproof.

###### Proof.

We prove the claim for the case that an agent is only allowed to create two identities. The claim holds in general, as we can alway merge two identities into one without decreasing the payoff, following the result of the case of two.

Fixing an arbitrary belief of agent , we denote the , where is the distribution specified by mechanism . Suppose an agent divides its wager into two wagers , and reports two predictions correspondingly. We have ,