Joint-outcome prediction markets for climate risks

their beliefs, that is until fj ¼ eqj =b Pm i¼1 eqi =b The exposure of the market maker that corresponds to these prices is given by qj ¼ b log fj þ b log X m i¼1 eqi =b ! ¼ b log fj þ bA where A ¼ log Pm i¼1 eqi =b �. Buying this exposure will cost the participant Cf ¼ b logðeA X m i¼1 fiÞ b log m ¼ bA b log m If outcome k is the one that ultimately occurs, then the participant will receive qk, so their net reward will be qk Cf ¼ b log fk þ b log m which is linear in log fk, which is the logarithmic scoring rule. The largest net reward the par- ticipant can receive, and hence the largest net loss the market maker can suffer, is bounded by the case fk = 1, and is given by b log m. This shows how the parameter b controls the market maker’s potential loss. The software platform allows participants to define contracts, w = (w1,� � �,wm), consisting of combinations of outcomes in the joint-outcome space and then trade these contracts via the mar- ket maker. Since contracts can be defined to cover every possible interval in one variable, partici- pants can take positions solely in temperature or rainfall. Participants can sell contracts they had previously bought at any time, although “shorting” (selling contracts that they do not own) is not permitted. However, because the pricing algorithm ensures the total price of the entire outcome space is always 1.00, if participants believe some outcomes are overpriced, they can exploit this by purchasing underpriced outcomes. Participants can trade using the platform’s user interface (UI)

or programmatically through an application programming interface (API). The UI only allows wi2{0,1}, whereas the API allows arbitrary non-negative values of wi to be used. The LMSR market maker rewards participants according to same scheme proposed by San- droni [13] when the proper scoring rule used is the logarithmic score, except that the “bench- mark” probabilities which participants must improve upon are provided by the prevailing market prices rather than the forecast user. Initialization auction If the market maker’s initial exposure is qi = 0 (i = 1,� � �,m) then initial prices are 1/m for all outcomes. This is typically unrealistic and, as soon as the market opens, participants will rush to snap up underpriced outcomes. To try and avoid this rush, the initial prices in the markets were determined using an auction. For one week before market opening participants could submit orders consisting of a contract definition, w = (w1,� � �,wm), the price the participant was prepared to pay per unit of that contract, R, and the maximum number of contracts they were prepared to buy at that price, xmax. Once the bidding was closed, an algorithm prioritized orders based on the premium offered above what the market maker would ask for in live trading. The maximum premium for an PLOS ONE Joint-outcome prediction markets for climate risks PLOS ONE | https://doi.org/10.1371/journal.pone.0309164 August 30, 2024 7 / 14