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Belief Aggregation with Automated Market Makers

Abstract

We consider the properties of a cost function based automated market maker aggregating the beliefs of risk-averse traders with finite budgets. Individuals can interact with the market maker an arbitrary number of times before the state of the world is revealed. We show that the resulting sequence of prices is convergent under general conditions, and explore the properties of the limiting price and trader portfolios. The limiting price cannot be expressed as a function of trader beliefs, since it is sensitive to the market maker’s cost function as well as the order in which traders interact with the market. For a range of trader preferences, however, we show numerically that the limiting price provides a good approximation to a weighted average of beliefs, inclusive of the market designer’s prior belief as reflected in the initial contract price. This average is computed by weighting trader beliefs by their respective budgets, and weighting the initial contract price by the market maker’s worst-case loss, implicit in the cost function. Since cost function parameters are chosen by the market designer, this allows for an inference regarding the budget-weighted average of trader beliefs.

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Notes

  1. 1.

    See, especially, Hayek (1945), who drew attention to the importance of the latter role.

  2. 2.

    Chen and Plott (2002) make this point as follows: “Gathering the bits and pieces by traditional means, such as business meetings, is highly inefficient because of a host of practical problems related to location, incentives, the insignificant amounts of information in any one place, and even the absence of a methodology for gathering it. Furthermore, business practices such a quotas and budget settings create incentives for individuals not to reveal their information.”

  3. 3.

    Among corporations, the list includes Microsoft, Intel, Eli Lily, GE, Siemens, and many others (Charette 2007; Broughton 2013). Providers of software for the implementation of prediction markets include Inkling, Consensus Point, and Lumenogic.

  4. 4.

    Chen and Plott (2002) report that the sum of the market prices of a set of binary securities on mutually exclusive and exhaustive events exceeded the amount that the single winning security would pay off in all 12 experiments in the HP market.

  5. 5.

    Abernethy et al. (2013) generalized the idea of a market scoring rule to settings in which the state space is exponentially large compared with the set of offered securities, and fully characterized the class of automated market makers that guarantee no arbitrage, bounded market maker loss, and other desirable properties.

  6. 6.

    To accommodate an unbounded number of trades one could assume, as in Ostrovsky (2012), that the sth trade occurs at time \(1-1/s\) and that the state is revealed at time 1. In practice, convergence to a limiting price is quite rapid and requires just a few rounds of trading.

  7. 7.

    For simplicity we assume all traders share the same utility function, but all theoretical results carry over to the setting in which each trader i has a distinct utility function \(u_i\) satisfying the criteria in Assumption 4.

  8. 8.

    Specifically, risk neutrality corresponds to \(\rho = 0\) and log utility to the limit as \(\rho \rightarrow 1\). Risk neutrality falls outside of our model as Assumption 4 is violated.

  9. 9.

    All proofs are in the Appendix.

  10. 10.

    We experimented with smaller convergence tolerances; the results change only negligibly.

  11. 11.

    Leaving out the market maker, the average absolute difference goes up to 0.0226 and the average squared difference to 0.00078.

  12. 12.

    We use this class of distributions because it is defined on the unit interval, includes the uniform as a special case, and allows for both single-peaked and bimodal density functions.

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Correspondence to Rajiv Sethi.

Appendix

Appendix

Proof of Lemma 1

To prove the first case, assume that \(p_i \le \pi _{t-1}\) and \(z_{i,t-1} \ge 0\). By the convexity of C, showing that \(\pi _{t} \le \pi _{t-1}\) is equivalent to showing that trader i does not select \(r_t > 0\). For this, it suffices to show that he prefers \(r_t = 0\) to any \(r_t > 0\).

Consider some \(r > 0\). By the convexity of C, the cost of purchasing r at time t is at least \(\pi _{t-1} r\) which is at least \(p_i r\) by our assumption that \(p_i \le \pi _{t-1}\). The expected utility of trader i after making this purchase is therefore upper bounded by the function

$$\begin{aligned} v(r) = p_i u(y_{i,t-1} + z_{i,t-1} - p_i r + r) + (1-p_i) u(y_{i,t-1} - p_i r) . \end{aligned}$$
(5)

Taking the derivative of this function with respect to r yields

$$\begin{aligned} v'(r) = p_i(1-p_i)\left( u'(y_{i,t-1} + z_{i,t-1} - p_i r + r) -u'(y_{i,t-1} - p_i r)\right) . \end{aligned}$$
(6)

Since u is concave (and so \(u'\) is decreasing) and \(z_{i,t-1} \ge 0, v'(r)\) is decreasing for \(r > 0\). Since v(0) is exactly the expected utility of trader i with \(r=0\) and v(r) is an upper bound on his utility when \(r>0\), this implies that trader i prefers \(r_t = 0\) to any value \(r_t > 0\), as desired.

The proof of the second case is analogous to the proof of the first. \(\square \)

Proof of Lemma 2

To prove the first case, assume that \(p_i \ge \pi _{t-1}\) and \(z_{i,t-1} \ge 0\). We must show that trader i does not select a bundle that would result in a price of \(\pi _{t} > p_i\). For this, it suffices to show that he would prefer to move the price to exactly \(p_i\) rather than to any higher price.

Since any cost function based market is path independent, moving the price from \(\pi _{t-1}\) to \(\pi _{t}\) costs the same as moving the price from \(\pi _{t-1}\) to \(p_i\) and then from \(p_i\) to \(\pi _{t}\). Therefore, it suffices to show that if trader i first moves the price to \(p_i\), he prefers to keep it there rather than subsequently moving it to a higher value.

By the convexity of C, the price can be increased from \(\pi _{t-1}\) to \(p_i\) only by purchasing a non-negative number of shares. Let \(\hat{z} \ge z_{i,t-1} \ge 0\) be the asset position of trader i after this move. Since his asset position is still non-negative, and the market price is now exactly equal to his beliefs, we can apply Lemma 1 to immediately show that he prefers to keep the price at \(p_i\) or decrease it rather than increase it, which completes the proof.

The proof of the second case is analogous to the proof of the first, relying on the second case in Lemma 1. \(\square \)

Proof of Proposition 1

Since \(\pi _0 = p_0\) and \(p_0 \in I\) by definition, it is clear that \(\pi _0 \in I\). Suppose, by way of contradiction, that there exists \(t \ge 1\) such that \(\pi _s \in I\) for all \(s < t\) and \(\pi _{t} \notin I\). We consider the case \(\pi _{t} > p_{\max }\) (the case \(\pi _{t} < p_{\min }\) may be proved analogously).

Let \(i = k(t)\), and suppose first that \(z_{i,t-1} = 0\). If \(p_i \le \pi _{t-1}\) then \(\pi _{t} \le \pi _{t-1} \le p_{\max }\) from Lemma 1 and the fact that \(\pi _{t-1} \in I\), a contradiction. And if \(p_i \ge \pi _{t-1}\) then \(\pi _{t} \le p_i \le p_{\max }\) from Lemma 2 and the fact that \(p_i \in I\), a contradiction.

Now suppose that \(z_{i,t-1} \ne 0\). Then there exists \(s < t\) such that \(k(s) = i\) and \(k(t') \ne i\) for all \(t' \in \{s+1,...,t-1\}\). We know that \((y_{i,t-1},z_{i,t-1})\) is an optimal portfolio for i at market state \(q_{s}\), when the market price is \(\pi _{s} \le p_{\max }\). If \(\pi _{t-1} = \pi _{s}\) then i will not change his portfolio in period t, so \(\pi _{t} = \pi _{t-1} \le p_{\max }\), a contradiction. If \(\pi _{t-1} > \pi _{s}\) then i will sell in period t, so \(\pi _{t} < \pi _{t-1} \le p_{\max }\), a contradiction.

Finally, if \(\pi _{t-1} < \pi _{s}\), then i will buy in period t. Any such transaction may be viewed as taking two steps in sequence: buy until the price reaches \(\pi _{s}\), and then buy or sell to reach the new optimum. After the first stage the market state will be \(q_{s}\) and the endowment will be (yz) where \(y < y_{i,t-1}\) and \(z > z_{i,t-1}\). Since i did not want to buy or sell at this market state with portfolio \((y_{i,t-1}, z_{i,t-1})\), he will want to sell with portfolio (yz). Hence \(\pi _{t} < \pi _{t-1} \le p_{\max }\), a contradiction. \(\square \)

Proof of Lemma 3

Suppose that trader i with endowment (yz) is considering purchasing r units of the asset at some market state q, and let \(c_q(r) = C(q+r) - C(q)\) denote the cost of this transaction. If \(r<0\), this is a sale and the cost is negative. The expected utility of trader i after this transaction is given by \( p_i u(y-c_q(r)+z+r) + (1-p_i) u(y-c_q(r)).\) Taking the derivative with respect to r gives

$$\begin{aligned} p_i (1-c'_q(r)) u'(y-c_q(r)+z+r) - (1-p_i) c'_q(r) u'(y-c_q(r)). \end{aligned}$$
(7)

Consider any t such that \(s(t) >0\) and let \(i = k(t)\). For notational simplicity, we will write s in place of s(t). We know that trader i would not want to buy or sell at endowment \((y_{i,s},z_{i,s})\) and market state q such that \(C'(q) = c'_q(0) = \pi _{s}\); otherwise, the path independence of the cost function implies that trader i would not have left the price in this state at time s. From Eq. 7, this tells us that

$$\begin{aligned} p_i (1-\pi _{s}) u'(y_{i,s}+z_{i,s}) - (1-p_i) \pi _{s} u'(y_{i,s}) = 0. \end{aligned}$$
(8)

Now consider the decision of trader i at time t. Since the endowment of trader i at the start of period t is precisely \((y_{i,s},z_{i,s})\) and the current price is \(\pi _{t-1}\), Eq. 7 tells us that trader i would want to buy a positive quantity of the asset if and only if \(p_i (1-\pi _{t-1}) u'(y_{i,s}+z_{i,s}) - (1-p_i) \pi _{t-1} u'(y_{i,s}) > 0.\) From Eq. 8, this holds if and only if \(\pi _{t-1} < \pi _s\). Similarly, trader i would want to sell a positive quantity (or buy a negative quantity) if and only if \(\pi _{t-1} > \pi _s\).

First consider the case in which \(\pi _{t-1} < \pi _s\), so the trader wants to buy. Suppose that i submits an order that restores the market to the state q such that \(C'(q) = c_q(0) = \pi _s\). Let \((y',z')\) denote the resulting endowment, and note that \(y' < y_{i,s}\) and \(y' + z' > y_{i,s} + z_{i,s}\). We shall show that at this endowment and price, the trader now wishes to sell. Consider a purchase (possibly negative) of r units starting from the endowment \((y',z')\) at market state q. As before, the expected utility is given by \(p_i u(y'-c_q(r)+z'+r) + (1-p_i) u(y'-c_q(r)),\) and its derivative at \(r=0\) is \( p_i (1-\pi _s) u'(y'+z') - (1-p_i) \pi _s u'(y').\) This must be less than 0 by Eq. 8, the concavity of u, and the fact that \(y' < y\) and \(y' + z' > y + z\). By path independence of the cost function, this implies that while i would like to buy at price \(\pi _{t-1}\), he would not buy enough to push the price back to \(\pi _s\), yielding the result. The proof for the case in which \(\pi _{t-1} > \pi _s\) is analogous. \(\square \)

Proof of Lemma 4

Let \(\Gamma = \{ (y,z) \, | \, y > 0, y+z >0 \},\) and define the function \(\psi : \Gamma \rightarrow (0,1)\) as

$$\begin{aligned} \psi (y,z) = {p_i u'(y+z) \over p_i u'(y+z) + (1-p_i) u'(y)}. \end{aligned}$$
(9)

From (8), the endowment \((y,z) \in \Gamma \) is optimal for trader i at price \(\psi (y,z)\) in the sense that a trader with portfolio (yz) would not want to buy or sell if the current price were \(\psi (y,z)\). \(\psi \) is continuous since u is smooth, so the inverse image \(\psi ^{-1}(E)\) of any closed set \(E \subset (0,1)\) is closed. In particular, \(\psi ^{-1}(\{ \pi \})\) is closed for any \(\pi \in I = [p_{\min }, p_{\max }]\).

Consider any t such that \(s(t) >0\), and let \(s = s(t)\) and \(i = k(t)\). Since \(\pi _s \in I\) and \(\lim _{w \rightarrow 0} u'(w) = \infty \), optimal portfolios will satisfy the non-negative wealth constraints with strict inequality in all periods. That is, \((y_{i,s},z_{i,s}) \in \Gamma \). By Lemma 3, the choice problem faced by trader i in period t with budget \(y = y_{i,s}\) and assets \(z = z_{i,s}\) may be expressed as follows: choose \(\alpha \in [0,1)\) to maximize

$$\begin{aligned} p_i u(y-c_{\pi _s, \pi _{t-1}}(\alpha )+z+r_{\pi _s, \pi _{t-1}}(\alpha )) + (1-p_i) u(y-c_{\pi _s, \pi _{t-1}}(\alpha )), \end{aligned}$$
(10)

where \(r_{\pi _s, \pi _{t-1}}(\alpha )\) is the (positive or negative) quantity of assets that trader i would need to purchase to bring the market price to \( \pi _t = \alpha \pi _{s} + (1-\alpha ) \pi _{t-1}\), and \(c_{\pi _s, \pi _{t-1}}(\alpha )\) is the cost of this purchase. The bounded loss of C implies that these quantities must exist since it must be possible to move the market price to anything in (0, 1) (Abernethy et al. 2013). Furthermore, one can easily verify that for any given values of \(\pi _s\) and \(\pi _{t-1}, r_{\pi _s, \pi _{t-1}}(\alpha )\) and \(c_{\pi _s, \pi _{t-1}}(\alpha )\) are both continuous since the cost function C is smooth and convex. The necessary and sufficient condition for a maximum is

$$\begin{aligned}&p_i (r'_{\pi _s, \pi _{t-1}}(\alpha )-c'_{\pi _s, \pi _{t-1}}(\alpha )) u'(y-c_{\pi _s, \pi _{t-1}}(\alpha )+z+r_{\pi _s, \pi _{t-1}}(\alpha )) \\&\quad -\, (1-p_i) c'_{\pi _s, \pi _{t-1}}(\alpha ) u'(y-c_{\pi _s, \pi _{t-1}}(\alpha )) = 0. \end{aligned}$$

For any given tuple \((y,z,\pi _s,\pi _{t-1})\) with \(\pi _s \ne \pi _{t-1}\), this condition implies a unique solution \(\alpha (y,z,\pi _s,\pi _{t-1})\) by Lemma 3. By the continuity of \(u(\cdot ), r_{\pi _s, \pi _{t-1}}(\cdot )\), and \(c_{\pi _s, \pi _{t-1}}(\cdot ), \alpha (\cdot )\) is also continuous where it is defined.

Note that for any \(\eta > 0, \alpha (\cdot )\) is well-defined on the domain

$$\begin{aligned}&\Delta = \{ (y,z,\pi _s,\pi _{t-1}) \in \mathbb {R}^4 \, | \, \\&\quad (\pi _s,\pi _{t-1}) \in [p_{\min },p_{\max }]^2, \, (y,z) \in \psi ^{-1} (\{ \pi _s \}), \, |\pi _s - \pi _{t-1} | \ge \eta \}. \end{aligned}$$

Since \(\psi ^{-1} (\{ \pi _s \})\) is closed and bounded, \(\Delta \) is compact. Since compactness is preserved by continuous functions, \(\alpha \) must also have a compact range, which excludes \(\alpha = 1\) by Lemma 3; although some states in \(\Delta \) may not be reachable in the market, the proof of Lemma 3 holds for all states in this set. Hence the range of \(\alpha \) over domain \(\Delta \) must have a maximum element \(\bar{\alpha }(\eta ) < 1\). \(\square \)

Proof of Lemma 6

Note that for any \(\eta >0\), there exist \(\varepsilon >0\) and \(\delta >0\) such that

$$\begin{aligned} \eta > \varepsilon + {\delta + \bar{\alpha }(\eta ) \varepsilon \over 1 - \bar{\alpha }(\eta )}, \end{aligned}$$
(11)

since \(\bar{\alpha }(\eta ) < 1\) from Lemma 4. Let \(\eta >0\) be given and consider any positive \(\varepsilon \) and \(\delta \) consistent with (11). By definition of \(\bar{\pi }\), there exists \(t'\) such that, for all \(t > t'-m, \pi _t < \bar{\pi } + \varepsilon \). Consider any \(\tau > t'\) with \(\pi _\tau > \bar{\pi } - \delta \). Clearly \(\pi _{s(\tau )} < \bar{\pi } + \varepsilon \). By Lemma 3,

$$\begin{aligned} \pi _\tau = \alpha _\tau \pi _{s(\tau )} + (1-\alpha _\tau ) \pi _{\tau -1}. \end{aligned}$$
(12)

This implies \( (1-\alpha _\tau ) \pi _{\tau -1} = \pi _\tau - \alpha _\tau \pi _{s(\tau )} > \bar{\pi } - \delta - \alpha _\tau (\bar{\pi } + \varepsilon ).\) Hence

$$\begin{aligned} \pi _{\tau -1} > \bar{\pi } - {\delta + \alpha _\tau \varepsilon \over 1-\alpha _\tau } > \pi _{\tau } - \left( \varepsilon + {\delta + \alpha _\tau \varepsilon \over 1-\alpha _\tau } \right) \end{aligned}$$
(13)

where the last inequality follows from the fact that \(\pi _{\tau } < \bar{\pi } + \varepsilon \).

We claim that \(\pi _{\tau -1} > \pi _\tau - \eta .\) Suppose not. Then \(\pi _\tau - \pi _{\tau -1} \ge \eta \), which by (12) implies that \(\pi _{s(\tau )} - \pi _{\tau -1} \ge \eta \), and \(\alpha _\tau \le \bar{\alpha }(\eta )\). Hence from (13), we obtain

$$\begin{aligned} \pi _{\tau -1} > \pi _{\tau } - \left( \varepsilon + {\delta + \bar{\alpha }(\eta ) \varepsilon \over 1-\bar{\alpha }(\eta )} \right) , \end{aligned}$$
(14)

which implies \( \pi _{\tau -1} > \pi _\tau - \eta \) from (11), a contradiction. Hence \( \pi _{\tau -1} > \pi _\tau - \eta .\) Note that \(\delta < \eta \) from (11), so \( \pi _{\tau -1} > \bar{\pi } - \delta - \eta > \bar{\pi } - 2\eta .\) Setting \(\eta = \gamma /2\) yields the desired result. \(\square \)

Proof of Theorem 1

From Lemma 6, for any \(\gamma >0\), there exists \(t' \in \mathbb {N}\) and a sequence of positive numbers \(\delta _1,...,\delta _{m}\) such that \(\gamma = \delta _1 > \delta _2 > ... > \delta _{m} > 0\) and, for all \(t > t'\) and \(i = 2,...,m, \pi _{t+i} > \bar{\pi } - \delta _i \implies \pi _{t+i-1} > \bar{\pi } - \delta _{i-1}.\) Furthermore, there exists \(t > t'\) such that \(\pi _{t+m} \ge \bar{\pi } > \bar{\pi } - \delta _m\). Suppose that \(\bar{\pi } > \underline{\pi }\) and set \(\gamma = \bar{\pi } - \underline{\pi }\). Then there exists a sequence of m consecutive prices \(\pi _{t+1},...,\pi _{t+m}\) all of which exceed \(\underline{\pi }\). Hence \(\underline{\pi }_{t+m} > \underline{\pi }\), a contradiction. \(\square \)

Proof of Proposition 2

Consider a trader with belief p, portfolio (yz), facing price \(\pi \). From the first order condition for optimality, this trader will choose to remain at this portfolio if and only if

$$\begin{aligned} p (1-\pi ) u'(y+z) = \pi (1-p) u'(y), \end{aligned}$$
(15)

or

$$\begin{aligned} \pi = { pu'(y+z) \over p u'(y+z) + (1-p) u'(y)} . \end{aligned}$$
(16)

Concavity of u implies that

$$\begin{aligned} \pi = { pu'(y+z) \over p u'(y+z) + (1-p) u'(y)} < p \end{aligned}$$
(17)

if and only if \(z>0\). Similarly, \(\pi > p\) if and only if \(z < 0\). Since the terminal portfolio is optimal given the terminal price, the result follows. \(\square \)

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Sethi, R., Vaughan, J.W. Belief Aggregation with Automated Market Makers. Comput Econ 48, 155–178 (2016). https://doi.org/10.1007/s10614-015-9514-7

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Keywords

  • Prediction markets
  • Automated market makers
  • Belief aggregation