5202 yaM 72 ]HT.noce[ 1v69212.5052:viXra Repeated Auctions with Speculators: Arbitrage Incentives and Forks in DAOs Nicolas Eschenbaum, Nicolas J. Greber Abstract. We analyze the vulnerability of decentralized autonomous organizations (DAOs) to speculative exploitation via their redemption mechanisms. Studying a game-theoretic model of repeated auctions for governanceshareswithspeculators,wecharacterizetheconditionsunder which—in equilibrium—an exploitative exit is guaranteed to occur, oc- cursinexpectation,orneveroccurs.Weevaluatefourredemptionmech- anisms and extend our model to include atomic exits, time delays, and DAO spending strategies. Our results highlight an inherent tension in DAO design: mechanisms intended to protect members from majority attacks can inadvertently create opportunities for costly speculative ex- ploitation. We highlight governance mechanisms that can be used to prevent speculation. Keywords: DecentralizedAutonomousOrganizations(DAOs),repeated auctions, treasury redemption, speculative bidding 1 Introduction Decentralized autonomous organizations (DAOs) are a novel governance struc- tureenabledbyblockchaintechnologythatallowsdecentralizeddecision-making and management of collective resources. A major concern for such decentralized organizations is the risk of “majority attacks,” where a group holding the ma- jorityofvotingrightsdecidestoexpropriatethecollectiveresources.Tomitigate this, many DAOs allow members to exit before the decision takes effect, ensur- ing they receive their share of the joint assets upon departure. However, this mechanism can unintentionally attract speculators who join solely to exit with a share of the treasury. In this paper, we study the incentives for speculators (or “arbitrageurs”) to participate in repeated auctions for individual shares of the organization in or- der to profitably exploit this mechanism. We focus on the case of the Nouns DAO—one of the largest and most influential DAOs where proportional claims to the DAOs treasury are auctioned daily through unique NFTs.1 In September 2023, the exit from the DAO by a substantial share of its members led to a loss 1 In May 2023, the treasury of the DAO held about $55 million, and the DAO has used its funds, for example, to fund a Super Bowl commercial with Bud Light in 2022 or name a rare species of frog in Ecuador. As of March 13, 2025 its treasury amountstoapproximately$6million.SeetheNounswebsiteforthecurrenttreasury and auction at https://nouns.wtf. 2 Nicolas Eschenbaum, Nicolas J. Greber of(atthetime)approximately$27million.Whileouranalysisiscenteredonthe Nouns DAO, the underlying economic mechanisms resemble a broader range of economic contexts, such as equity crowdfunding platforms with auction mecha- nisms or initial public offerings (IPOs) where participants similarly invest funds to acquire voting rights and proportional claims to organizational resources. We study a game-theoretic model of a repeated English auction with two types of bidders: regular members (or “Nouners”) who value their share (i.e., NFT)withsomepositivevaluationandspeculators(“Arbitrageurs”)whoseval- uation depends on the treasury size, conditions under which they may exit and redeemtheirshare,andtheresultingexpectedtimeuntilredemption.2 Weshow that there are three types of equilibria depending on the likelihood of an exit— with an exit occurring with certainty, an exit occurring in expectation, or no exit with certainty, respectively—and characterize the conditions under which theyarise.Wethenproceedtostudyfourdifferentexitmechanismsthatchange the share of the treasury received by members exiting and various extensions of the model that allow the organization to make exits less likely or prevent them altogether. In particular, we show how either commitment to a spending path or capping share values at the initial purchase price necessarily prevents partic- ipation by speculators.3 We provide numerical simulations of our main findings thatdemonstratehowthedifferentmechanismsandparametervaluesdetermine the equilibrium type. Our work is closely related to the literature on auctions with speculators. Bikhchandani and Huang in [1] consider auctions where all bidders are pure intermediariesbiddingsolelyforresale,asintreasurybillmarkets.Haileprovides in [3] empirical evidence from timber auctions showing that the option to resell addsvalueforbiddersandcaninjectacommon-valueelementeveninaprivate- value environment. Similar to our case, Garret and Tr¨oger (see [2]) show how a bidder with zero value who can profit by strategically bidding to win and then reselltoahigher-valuebidderintroducesmultipleequilibriatothegame:besides thehonest“bid-your-value”outcome,thereexistspeculativeequilibriawherethe speculatorwinsatalowpriceandlatersellstotherealuser.4 Incontrasttothis literature, we do not model an option to resell the share, but instead, we allow winners to redeem their share of the organization’s assets in cash value. Ouranalysisisalsorelatedtotheliteratureoninitialpublicofferings(IPOs) and equity crowdfunding auctions. Many IPO participants are “flippers” who buy at the offering and resell quickly (“flipping” the stock). Similarly, Lukkari- nen and Schwienbacher (see [4]) show that simply announcing a plan to enable 2 In our model, we require a threshold number of shares to be held by speculators to force an exit. 3 Note that commitment is non-trivial to ensure for a decentralized organization and cappingNFTvaluesistechnicallychallengingandaddspropertiestotheNFTsthat affect secondary markets. 4 Pagnozzishowsin[5]that,infact,high-valuebiddersmaysometimespreferaspec- ulator to win. Repeated Auctions with Speculators 3 secondarytradingcansignificantlyboostinvestorparticipationinequitycrowd- funding. This paper is structured as follows. Section 2 develops the analytical frame- work. Section 3 characterizes the behavior of bidders in equilibrium. Section 4 provides our main results on the types of equilibria and conditions under which they arise. Section 5 develops various extensions of our model and section 6 concludes. All proofs are relegated to the appendix. 2 Model Setup Let S denote the treasury stock at time t∈1,...,T, with T >1, where S ≥0 t 0 isthetreasurypriortothegamecommencing.Thenumberofnounsatthestart of the game is given by N >0 which are all held by nouners. At each time t one new noun is auctioned off. Nouns serve as voting rights in the Nouns DAO. We denote by p the realized auction price at time t. The treasury stock therefore t evolves over time according to S =S + (cid:80)τ=t−1p . t 0 τ=1 τ Nounscanberedeemedforashareα ofthetreasury.Wedenotetheexpected t value of one noun at time t conditional on the history of the game h and for an t expected redemption period t∗ ≥ t by Ve(h ), where we suppress the subscript t t of the expected redemption period t∗ for brevity. We will refer to Ve(h ) as the t t redemption value. In the special case of t∗ = t, we write the redemption value simply as Ve (h )=V(h ). t=t∗ t t Allplayersdiscountfuturepayoffsbythesamediscountfactorδ ∈(0,1).The expected redemption value is thus given by Ve(h ) = δt∗−tE[α S (h )], where, t t t t t as before, we suppress the subscript t∗ for brevity. We assume that nouners value each noun according to a valuation v˜∼[0,v¯], with v¯ > 0 constant over time. Let F denote the distribution function for v˜. We assume that F has a density f that is positive and continuous on [0,v¯] and is identically 0 elsewhere. This represents the natural demand for nouns. Note that throughout, we will denote random variables with a tilde, e.g., v˜, and their realizations without, e.g., v. Arbitrageurs (or speculators), in turn, value the noun at the expected value of redemption Ve(h ). In other words, we make a sharp distinction between t t nouners—whovaluenounsindependentoftheirredemptionvalueoranypossible marketvalueonasecondarymarket—andarbitrageurs,whoonlyvaluethemfor theredemptionvalue.Nounscanonlyberedeemed,however,afteraforkoccurs. Weassumethatonlyarbitrageurswanttoforkand,whentheydo,willinstantly redeem their nouns. For a fork to occur at time t, the number of nouns held by arbitrageurs, A , must exceed a constant forking threshold κ ∈ (0,1), given by t N A + t t ≥κ. At the start of the game, no arbitrageur holds a noun or A 0 =0. Thetimingofthemodelisasfollows.Ateachtimet,theauctioncommences with one arbitrageur and n≥2 nouners participating. The auction format is an English auction with a starting price of 0. Following the auction, the realized auction price is added to the treasury, and a fork occurs if A ≥ κ(N +t). If it t does, then all arbitrageurs fork and redeem their nouns, and the game ends. 4 Nicolas Eschenbaum, Nicolas J. Greber We model the auction format as an English auction, as this most closely resembles the auctions for nouns, in which bids can be submitted over a 24- hour period and all bids are visible at all times. Ending the game after the fork is WLOG, as all arbitrageurs leave the DAO and redeem their nouns. The continuation game is, therefore, simply a repetition of the game with a new starting treasury value S and a number of nouns N. We assume that multiple 0 nouners participate in order to guarantee that the expected auction price is always strictly positive. Thehistoryh containstreasurystocksandplayers’bidsforallpastperiods. t However, it is straightforward that the current treasury stock, number of nouns held by arbitrageurs at the beginning of the period, and sum of past prices containallinformationrequiredforplayers’strategies.5 Letthesumofpricesbe denoted by P = (cid:80)t−1 p . Then it is sufficient for the sequence of histories t−1 τ=1 τ to be defined as h ={S ,A ,P }, h ={S ,A ,P ,S ,A ,P }, etc. The set of 1 0 0 0 2 0 0 0 1 1 1 histories is denoted by H . t Let˜b (·|h )denotethe(possiblyrandomized)bidatwhichnouneri∈1,...,n i,t t planstodropoutathistoryh asafunctionofherrealizedvaluev andsimilarly t i,t let˜b (·|h ) denote the arbitrageurs bid. We further denote by˜b the highest a,t t −i,t of the maximum bids of all players other than i and by ˜b the highest of the −a,t maximum bids of all nouners. For all bids ˜b ≥ 0 and v ∈ [0,v¯], nouner i’s i,t i,t expected payoff at time t is then given by (cid:104) (cid:105)(cid:16) (cid:17) u (b ,v )=Prob ˜b ≥˜b | h v −˜b . (1) i,t i,t i,t i,t −i,t t i,t −i,t Similarly,forallbids˜b ≥0,thearbitrageur’sexpectedpayoffattimetisgiven a,t by (cid:104) (cid:105)(cid:16) (cid:17) u (b ,h )=Prob ˜b >˜b | h Ve(h )−˜b . (2) a,t a,t t a,t −a,t t t t −a,t Ties between nouners are broken at random6. 3 Preliminaries Webeginouranalysisbystudyingthebiddingbehaviorofplayersatanyhistory h . t Lemma 1. (Bidding Strategies) In any equilibrium and at any h , t i. all nouners i ∈ 1,...,n always bid up to their valuation of the noun, or ˜b =v , i,t i,t ii. allarbitrageursalwaysbiduptoacutoffvalueˆb (h )≥min{v ,Ve(h )}, t t n−1,t t t or ˜b (h )=ˆb (h )≥min{v ,Ve(h )}, a,t t t t n−1,t t t 5 Dependingonthespecificationoftheshare,thesumofpastpricesneednotbepart of the history. This is the case, for example, under the current pro-rata mechanic (see Section 4) 6 This assumption is innocuous in equilibrium, as the analysis below shows. Repeated Auctions with Speculators 5 where v denotes the second-highest value drawn amongnounerns i=1,...,n n−1,t at history h . t Proof. in the appendix. Lemma 1 shows that the equilibrium of the game consists of a pair of simple strategies for nouners and the arbitrageur. The first statement formalizes the standard result that it is optimal for players in an English auction to bid up to their true valuation. However, this result only directly applies to nouners, becauseanarbitrageur’svaluation,theredemptionvalue,isafunctionofherown bidandexpectedfuturebidsand,thus,isdeterminedinequilibrium.Thesecond statement then shows that while pinning down arbitrageurs’ bids in equilibrium ismorecomplex,theiroptimalstrategiesstillfollowasimplestructureofbidding up to a well-defined maximum bidˆb (h ) that is at least as high as the lower of t t the arbitrageurs’ redemption value and the second-highest nouner valuation. Since all nouners always bid up to their true valuation of the noun, we can now state the expected time to fork at a given period t and history h as a t function of nouners valuations since the probability of arbitrageurs winning in expectation simply depends on the realizations of the highest nouner valuation v . n,t Lemma 2. (Forking): In any equilibrium and at any h , a necessary condition t for t∗ to exist in expectation is given by t∗ A + (cid:88)(cid:16)(cid:104) F(ˆb ) (cid:105)n(cid:17) ≥κ(N +t∗). (3) t τ τ=t Proof. in the appendix. Lemma2showsthattheexpectedtimetofork(andtheexistenceofaforking periodatall)isdrivenbythedistributionofnouners’valuationsandtheexpected redemption value of arbitrageurs. The higher the redemption value, the more likelyitisforthearbitrageurtowinsincearbitrageurs’willalwaysatleastoutbid nounerswheneverpossiblebyLemma1.Butanarbitrageur’swininexpectation, therefore, only depends on the probability of their expected redemption value being above the highest nouner valuation; thus, they are simply a function of the distribution function of nouners’ valuations. Specifically, the probability of winning conditional on the expected redemp- tion value and resulting optimal maximum bid, F(ˆb ), must be greater than κ t forthegametomoveclosertoaforkoccurringinexpectation.Ifitdoes,then,in expectation,arbitrageurs’winsaccumulatefasterthanthe’cost’ofalargernum- berofperiodshavingbeenplayedand,thus,therequirednumberofarbitrageur wins increasing (as a function of κ). But note that Lemma 2 only provides a necessary condition. It does not guarantee that a fork will occur. The critical functionoftheconditioninLemma2andthefactthatitonlyguaranteesafork inexpectationisillustratedinFigure1.Notethatthedisplayedevolutionofthe expected number of arbitrageur wins, E[A ], is illustrative only. t 6 Nicolas Eschenbaum, Nicolas J. Greber Fig.1: The relationship between the expected number of arbitrageur wins and the increasing number of nouns required to reach a fork. Our final preliminary result now provides an explicit characterization of the optimal bidding behavior of arbitrageurs in equilibrium and shows how their behavior is determined by the effect an arbitrageur’s bid has on the evolution of the redemption value. However, this only applies to interior solutions for the optimal maximum bid, denoted by b∗(h ). There can be equilibria in which an t t arbitrageur’s maximum bid lies outside the range of nouners valuations. Intu- itively,thiswould,forexample,bethecaseiftheinitialtreasuryissolargethat theexpectedvalueofashareofitishigherthananynouneriswillingtopayfor a noun. Lemma 3. (Arbitrageur Bid): In any equilibrium and at any h , if the optimal t maximal bid by an arbitrageur satisfies b∗(h )∈[0,v¯], then t t 1F(ˆb (h )) ∂Ve(h ) b∗(h )=Ve(h )+ a,t t t t . (4) t t t t n f(ˆb (h )) ∂ˆb (h ) a,t t a,t t Proof. in the appendix. Lemma 3 shows that arbitrageurs optimize their bid relative to the redemp- tion value. If the redemption value increases in their bid, then arbitrageurs may “overbid”—that is, holding the redemption value fix for a given bid, they op- timally bid more than this redemption value. If instead the redemption value decreases in their bid, then they would underbid. Whether arbitrageurs over- or underbid, therefore, depends on how the redemption value changes in their bid and the value of the hazard rate of nouners distribution at this bid level. As Lemma 1 shows however, underbidding cannot arise in equilibrium since the optimal strategies of nouners are independent of arbitrageurs’ strategies imply- ing that while an arbitrageurs bid may affect the auction price (and, of course, may lead to him winning the auction), an increasing bid can never lead to a decrease in the auction price as this would require other players to react to the arbitrageurs’ strategy. Repeated Auctions with Speculators 7 4 Main Results We now provide our main results on the equilibrium outcome of the game. Our firstkeyfindingstudiestheconditionsunderwhichaforkoccursinequilibrium. Proposition 1. (Equilibrium)Theequilibriumofthegameisoneofthreetypes, which are characterized by the following conditions: Type I: a fork is guaranteed to occur along the equilibrium path and the op- timal maximum bid by arbitrageurs satisfies b∗(h )≥v¯ if T ≥ κ N t t 1−κ and S ≥ v¯ , 0 δT−tα Type II: a fork is expected to occur along the equilibrium path starting at history h and the optimal maximum bid by arbitrageurs satisfies t b∗(h )≥Ve(h ) and b∗(h )∈(0,v¯) if and only if t t t t t t t∗ (cid:88) [F (b∗(h )×g(α,τ,h ))]n ≥κ(N +T)−A (5) t t t t τ=t for any t∗ ≤T, Type III: no fork is expected to occur along the equilibrium path starting at history h and the optimal maximum bid by arbitrageurs satisfies t b∗(h )=0 if and only if t t t∗ (cid:88) [F (b∗(h )×g(α,τ,h ))]n <κ(N +T)−A (6) t t t t τ=t for all t∗ ≤T, where g(·) is given by g(α,τ,h t )= δτ 1 −t α α τ t a τ,t , and a τ,t = (cid:34) (St+ (cid:16) S (cid:80) t+ t τ ∗ = (cid:80) τ t τ E ∗ = [p t τ E ] [ ) p + τ] n 1 (cid:17) + F f( ( n 1 E E [ [ b b F f ∗ τ ∗ τ ( ( ] b ] b ) ) ∗ t ∗ t ) (cid:80) )(cid:80) t τ ∗ = t τ ∗ τ = ( t n ( F nF (E ( [ b b ∗ t ∗ τ ) ] n )n − − 1( 1 1 ( − 1− F F (b ( ∗ t E ) [ ) b ) ∗ τ ]))) (cid:35) (7a) ≥0. (7b) Furthermore, a sufficient condition for a Type II equilibrium is given by t∗ (cid:88) [F (Ve(h )×g(α,τ,h ))]n ≥κ(N +T)−A for any t∗ ≤T. (8) t t t t τ=t Proof. in the appendix. Proposition1characterizesthethreepossibletypesofequilibriaandprovides conditionsunderwhichthegamewillendineachtypeofequilibrium.Intuitively, the result of the game depends on whether arbitrageurs’ optimal bids lie within theintervalofthedistributionofnounervaluations.Iftheydoandaresufficiently high, then the accumulating expected wins by arbitrageurs can be high enough 8 Nicolas Eschenbaum, Nicolas J. Greber forarbitrageurstoexpectaforktooccurandbewillingtobidapositiveamount (TypeII).Iftheyarenotsufficientlyhigh,theninexpectation,noforkwilloccur before the game ends, so that arbitrageurs are not willing to bid any positive amount(TypeIII).Finally,iftheylieoutsidetheboundsofthedistribution,then arbitrageurswillalwayswin,andaslongasthegamelastslongenoughforthem to surpass the forking threshold, a fork is guaranteed (Type I). However, note that our conditions for Type I only characterize the case in which the initial treasury stock S is so large that a Type I equilibrium arises. There can be 0 intermediateparametervaluesthatalsoresultinTypeIequilibriabeingplayed, which are not captured by our conditions. We also note that our conditions are not precise restrictions on the model primitives that must be fulfilled for the optimal bid by arbitrageurs to result in the respective type of equilibrium, but rather are expressed in terms of the optimal arbitrageur bid and thus the redemption value (which is a function of the model primitives), because the expected time to fork based on the sequence of win probabilities cannot be solved analytically. However, the conditions show that in order to characterize the equilibrium outcome of the model, it is suf- ficient to check whether, at the beginning of the game, arbitrageurs expect a fork to occur before the game ends. Moreover, to specifically check for a Type III equilibrium, the sufficient condition we provide, which is only a function of the redemption value rather than the full optimal bid by arbitrageurs, can be calculated instead. Figure 2 illustrates the equilibrium mechanics of the game. The expected optimal bids in future periods are increasing in line with the discount factor, starting with the optimal bid in period 1 (left panel), resulting in an increasing probability of winning (right panel). The total accumulated win probabilities of arbitrageurs (shaded area) represent the expected number of arbitrageur wins, and if this is sufficiently high, then a fork is expected to occur. In this case, the positive bids by arbitrageurs are indeed equilibrium play, and we are in a Type II equilibrium. The same logic holds true starting at any other period t instead of period 1 or for a forking period that is expected to occur before time T. In a Type III equilibrium, we instead observe an optimal arbitrageur bid of zero and the corresponding probability of winning of zero, while in a Type I equi- librium, the optimal bid lies above the bound of nouners valuations v¯ (and so does each future optimal bid) and thus the winning probability is always equal to one. Note that we display the bids and accumulating probability with con- tinuous functions while—in our model—time is discrete. Lastly, consider that the conditions stated in Proposition 1 hold for various possible assumptions rel- evant for the redemption value. In particular, we have not specified the share arbitrageurs receive when redeeming a noun. The conditions stated in Proposi- tion 1 only assume that the share arbitrageurs receive is not decreasing in their bid, that is, ∂α /∂b∗(h ) ≥ 0. We now proceed to numerically solve the model t∗ t t under different assumptions on the share arbitrageurs receive when redeeming a noun. We begin by imposing the current mechanism in the DAO on the model, specifically: Repeated Auctions with Speculators 9 Fig.2: Example of expected optimal future bids at period 1 and resulting ex- pectedaccumulatingarbitrageurwinsovertimeforthecaseofnounersvaluations being distributed according to a normal distribution. Mechanism 1 (Pro-rata share): α = 1 . t N+t When redeeming a noun, players receive the pro rata share of the entire treasury.Table1intheappendixshowsforthecaseofMechanism1howdifferent valuesfortheforkingthresholdκandtheinitialtreasuryS affecttheoutcomeof 0 the game for one particular set of parameter values and distribution function of nouners.Asexpected,lowerforkingthresholdsorhigherinitialtreasuriesreduce the redemption value and, hence, the probability of a fork and expected time to fork. In this setting, all three types of equilibria are possible, and even with an initialtreasuryofzero,aforkcanstilloccur.Thisisbecauseitcanbeprofitable for arbitrageurs to play relatively low but positive maximum bids, resulting in nouners winning in expectation for a number of rounds and paying their prices into the treasury. Once arbitrageurs are able to force a fork, the nouners’ paid prices result in a net gain for arbitrageurs. However, the forking threshold (κ) must be quite low for this to be an equilibrium strategy. The simulation runs for Table 1 in the appendix were conducted with the following parameters: the maximum number of periods was set to T =30, with the number of initial nouners N = 10 and a discount factor δ = 0.95. The number of nouners participating in the auctions was set at n=2 and the upper bound of the uniform distribution at v¯=1. The second assumption on the share arbitrageurs receive when redeeming the noun that we study is the following: Mechanism 2 (Pro-rata share with tax): α =c× 1 with c∈(0,1). t N+t In this case, arbitrageurs only receive a fraction of their pro-rata share. The remaining sum either stays in the DAOs treasury or is burned. It is straight- forward that all results from the pro-rata mechanism continue to apply, as it is simply the special case of Mechanism 2 for the case that c = 1. If the remain- ing sum stays in the DAOs treasury, our analysis is exactly identical to before. 10 Nicolas Eschenbaum, Nicolas J. Greber Whereas if the remaining sum is burnt, then all equilibrium mechanics continue to remain the same, but the starting initial treasury following the fork is lower. Tables 2a and 2b in the appendix document the numerical results for Mech- anism 2. The effect of setting a tax is similar to the effect of a higher κ or lower initial treasury: it reduces the redemption value and, thereby, the willingness to pay by arbitrageurs. In our simulations, the tax parameter (c) was set to 0.75 and 0.5, equivalent to a 25% and 50% tax, respectively. Asthetaxrateincreases,equilibriumconditionsaremetatlowerinitialtrea- surylevelsorhigherforkingthresholds.Loweringcmakesforkinglesslikelyand extends the time to fork. This effect is less pronounced than varying the initial treasury because even a smaller share of a positive treasury remains valuable, whereaschangingtheinitialtreasurysizemoredirectlyimpactstheprofitability of bidding for arbitrageurs. The third forking mechanism that we consider is the following, where we denote by j ≤t the time period in which a noun was bought: Mechanism 3 (Contribution-based share 1): α (h )= pj . j,t t (cid:80)t τ=1 pτ In contrast to the previous two mechanisms, under Mechanism 3 the share received after a fork does not depend on the number of nouns. Instead, it is a functionofthepricepaidforthenounorthecontributiontheownerofthenoun has made to the size of the treasury. Tables 3a and 3b document the simulation results for Mechanism 3. Com- paredtothepro-ratamechanism,Mechanism3leadstomore”extreme”equilib- ria, that is, Type II equilibria become very unlikely and in almost all scenarios, arbitrageurs either are guaranteed to win or do not participate in the auction (i.e., Type I and Type III equilibria). The reason for this finding is that under Mechanism 3, arbitrageurs are guaranteed to always obtain at least the price they paid. So they may as well bid high enough to guarantee themselves a win, whenever they can obtain a positive payoff by forking. However, they will only obtain this price at some future forking period t∗. Thus, due to discounting, they will value this return slightly less than the price they paid and Type II equilibria will become possible again. But this can only be the case under very specific parameter combinations, making Type II equilibria very rare. The final mechanism that we consider is the following: (cid:110) (cid:111) Mechanism 4 (Contribution-based share 2): α (h )=min pj, pj . j,t t St (cid:80)t τ=1 pτ ComparedtoMechanism3,Mechanism4putsacaponthesharearbitrageurs may receive. Just like in Mechanism 3, arbitrageurs receive a share proportional to the auction price they paid relative to the sum of auction prices, p /P . But j t whenever this share exceeds the ratio of their paid auction price to the total value of the treasury, they obtain this second share instead. Clearly, this would be the case when the initial treasury is non-empty. Figure 3 in the appendix illustrates the logic of this mechanism. The re- demption value for arbitrageurs does not continuously grow with the size of the Repeated Auctions with Speculators 11 treasury, instead it is capped at the level of their paid auction price relative to the overall size of the treasury. The consequence of this mechanism is stated formally in the next result. Corollary 1. (Mechanism 4): Assume Mechanism 4. Then, no Type I or Type II equilibria can arise and no forking occurs. Proof. in the appendix. 5 Extensions 5.1 Atomic Exit Insteadofplayersbeingonlyabletoforkonceasufficientnumberofnounholders choose to fork simultaneously, players can be allowed to exit individually at any point in time (an ’atomic exit’). Our model can capture this mechanism by placing two assumptions on the framework. First, we assume that κ = 0. As a consequence, the forking condition in Lemma 2 is no longer required, as it will alwaysbefulfilledonceanarbitrageurwinstheirauction.Second,theredemption value for arbitrageurs simplifies to Ve(h )=V(h )=α (S +E[p ]), t t t t t−1 t as the arbitrageur can fork after winning at time t immediately, and thus, her valueforthenounnolongerdependsonfutureauctionpricesorexpectedfuture arbitrageur wins. We can then state the following result, where we also charac- terize the development of the treasury as forks occur (i.e. if the game does not end after the first abers’ exit). Proposition 2. (Atomic exit) Assume that κ=0 so that players may fork and redeem their noun at any time t. Then, in any equilibrium and at any history h , a fork (atomic exit) occurs in expectation if t b∗(h )∈[0,v¯] and F (V(h ))>F(E[v ]), (9a) t t n,t and the expected value of the treasury follows a mean-reverting process denoted by S˜ and defined by t S˜ ∼D(µ ,S,S¯), (10a) t t where 1 µ =(1−α F(b∗(h ))n)(µ +E[p ]), S >0, S¯< v¯. (11) t t t t t α t Proof. in the appendix. As Proposition 2 shows, in a steady state, the treasury will follow a treasury pathdefinedbytheexitmechanismandtheexpectedvalueofthesecondhighest nouner. Any treasury values that exceed this path will immediately result (in expectation)inanarbitrageurbuyinganountoexitwithprofit.Theconsequence of this change to the forking mechanic in the DAO is particularly stark under Mechanism 3 and 4, as the following statement shows. 12 Nicolas Eschenbaum, Nicolas J. Greber Corollary 2. (Arbitrage under atomic exit) At any time t and history h , in t any equilibrium: – under Mechanism 3, a fork occurs with certainty whenever S >P . t−1 t−1 – under Mechanism 4, no fork occurs. Proof. in the appendix. With the atomic exit mechanic, arbitrageurs no longer need to consider fu- ture expected arbitrageur wins. As a consequence, because they are guaranteed under Mechanism 3 and 4 to always obtain at least the price they paid in the auction, they will simply pay enough to be guaranteed to win the auction and fork immediately, whenever it yields a strictly positive payoff. This is the case underMechanism3andapositiveinitialtreasury.UnderMechanism4,however, thearbitrageurs’returnsarecappedandcanneverbestrictlypositive,andthus, we obtain the finding from Corollary 1 again that arbitrageurs will refrain from participating in the auction altogether. 5.2 Time Delay When players choose to fork and redeem their noun, the share of funds they receive can be delayed. The implementation of a vesting period significantly influencesthewillingnessofarbitrageurstoparticipate.Asthedelayinreceiving funds increases, the willingness of arbitrageurs to participate decreases. This vesting period can be represented with a parameter ∆ ∈ 1,2,...,T − 1 that represents the number of periods (i.e., days) during which the funds are locked. The redemption value then becomes: (cid:32) t∗ (cid:33) Ve(h )=δt∗−t+∆α S + (cid:88) E[p ] . (12) t t t∗ t−1 τ τ=t The main consequence of this change to the forking mechanism is that it reduces the immediate financial incentive for arbitrageurs due to the delayed access to funds. It is straightforward that the results from Proposition 1 carry over to this scenario, as stated formally in the next result. Corollary 3. (Vesting): Assume funds received from redeeming a noun are ves- ted for ∆ periods. Then, the conditions for a Type II and Type III equilibrium from Proposition 1 continue to apply. The condition for a Type I equilibrium becomes: Type I: a fork is guaranteed to occur along the equilibrium path and the op- timal maximum bid by arbitrageurs satisfies b∗(h )≥v¯ if T ≥ κ N t t 1−κ and S ≥ v¯ . 0 δT−t+∆α Intuitively, because our conditions for a Type II and Type III equilibrium are a function of the optimal bid and thus the expected redemption value, their formaldefinitiondoesnotchange,butthesetofparametersforwhichtheyapply changes. Similarly, our condition for a Type I equilibrium now requires a larger initial treasury to guarantee a fork.

12 Nicolas Eschenbaum, Nicolas J. Greber Corollary 2. (Arbitrage under atomic exit) At any time t and history h , in t any equilibrium: – under Mechanism 3, a fork occurs with certainty whenever S >P . t−1 t−1 – under Mechanism 4, no fork occurs. Proof. in the appendix. With the atomic exit mechanic, arbitrageurs no longer need to consider fu- ture expected arbitrageur wins. As a consequence, because they are guaranteed under Mechanism 3 and 4 to always obtain at least the price they paid in the auction, they will simply pay enough to be guaranteed to win the auction and fork immediately, whenever it yields a strictly positive payoff. This is the case underMechanism3andapositiveinitialtreasury.UnderMechanism4,however, thearbitrageurs’returnsarecappedandcanneverbestrictlypositive,andthus, we obtain the finding from Corollary 1 again that arbitrageurs will refrain from participating in the auction altogether. 5.2 Time Delay When players choose to fork and redeem their noun, the share of funds they receive can be delayed. The implementation of a vesting period significantly influencesthewillingnessofarbitrageurstoparticipate.Asthedelayinreceiving funds increases, the willingness of arbitrageurs to participate decreases. This vesting period can be represented with a parameter ∆ ∈ 1,2,...,T − 1 that represents the number of periods (i.e., days) during which the funds are locked. The redemption value then becomes: (cid:32) t∗ (cid:33) Ve(h )=δt∗−t+∆α S + (cid:88) E[p ] . (12) t t t∗ t−1 τ τ=t The main consequence of this change to the forking mechanism is that it reduces the immediate financial incentive for arbitrageurs due to the delayed access to funds. It is straightforward that the results from Proposition 1 carry over to this scenario, as stated formally in the next result. Corollary 3. (Vesting): Assume funds received from redeeming a noun are ves- ted for ∆ periods. Then, the conditions for a Type II and Type III equilibrium from Proposition 1 continue to apply. The condition for a Type I equilibrium becomes: Type I: a fork is guaranteed to occur along the equilibrium path and the op- timal maximum bid by arbitrageurs satisfies b∗(h )≥v¯ if T ≥ κ N t t 1−κ and S ≥ v¯ . 0 δT−t+∆α Intuitively, because our conditions for a Type II and Type III equilibrium are a function of the optimal bid and thus the expected redemption value, their formaldefinitiondoesnotchange,butthesetofparametersforwhichtheyapply changes. Similarly, our condition for a Type I equilibrium now requires a larger initial treasury to guarantee a fork. Repeated Auctions with Speculators 13 5.3 Treasury Spending In our baseline model, all outflows from the treasury arise from players fork- ing and ragequitting the new DAO. In practice, funds from the treasury are regularly spent on proposals. Intuitively, introducing spending into the model will make forks less likely as the treasury shrinks and incentives to participate for arbitrageurs are reduced. Fully modeling the voting mechanism that decides on spending is beyond the scope of this analysis. However, we now introduce a reduced-form spending function into our setting and analyze its effects. Letz (h )denotetreasuryspendingattimetandhistoryh .Wewillassume t t t that the level of treasury spending is only a function of the current treasury stock at the beginning of the period so that z (h )=z (S ). (13) t t t t−1 The treasury stock, therefore, evolves over time according to S =S +p −z (S ). (14) t t−1 t t t−1 We further assume that the timing in each period is such that the auction price is added to the treasury first, then treasury spending occurs, and finally, a fork can arise. It is straightforward then that all our main results continue to hold but that the redemption value for arbitrageurs now needs to be adjusted for the expected level of treasury spending or (cid:32) t∗ (cid:33) Ve(h )=δt∗−tα S + (cid:88) (E[p ]−E[z (S )]) . (15) t t t∗ t−1 τ τ τ−1 τ=t It is immediately clear that depending on the expected level of spending, forks can be prevented entirely by, for example, committing to spend the entire trea- sury in each period. More generally, if the DAO were to commit to a spending pathovertime,thesameresultcanbeobtaineddespitenotspendingeverything at once. Specifically, suppose that the DAO commits to a declining spending path that takes the form of an exponential decay, or z (S )=kS e−λ(t−1), (16) t t−1 t−1 where 0<k <1 is a constant determining the fraction of the treasury spent in eachperiodandλ>0isthedecayrate.Then,wecanstatethefollowingresult. Proposition 3. (Treasury Spending): For any given value of S , there exists a 0 set of critical parameters kˆ, λˆ, such that no Type I or Type II equilibrium can arise. Proof. in the appendix. Proposition 3 shows that by choosing an appropriate spending path, incen- tives for arbitrageurs can be sufficiently reduced to prevent forking. Intuitively, 14 Nicolas Eschenbaum, Nicolas J. Greber by spending sufficiently quickly, arbitrageurs’ payoffs by the time they have ac- cumulated enough wins to force a fork are reduced far enough so that their optimalbids become low enough toslow downthe speedatwhichtheyaccumu- late wins in expectation to ensure the forking condition cannot be satisfied. As a consequence, even though they could obtain a positive payoff if a fork would arise,theycannevergetthere,soitnolongerpaystoparticipateintheauction, and only Type III equilibria remain. A similar set of parameters could also be determined at which only Type I equilibria are excluded. 6 Conclusion This paper studies the incentives for speculators to participate in repeated auc- tions for shares of the assets of a decentralized organization. Redemption mech- anisms that allow the winner of the auction to later redeem their share for cash value incentivize speculators to enter bids with the sole objective of exploit- ing the redemption mechanism. We build a game-theoretic model inspired by the Nouns DAO and characterize equilibrium bidding behavior and conditions under which successful arbitrage occurs. We show that there are three types of equilibria. In a Type I equilibrium, an exit by speculators is guaranteed along the equilibrium path because the specu- lators’optimalmaximumbidexceedsthehighestvaluationofregularbidders.In aTypeIIequilibrium,aforkoccursinexpectation,andthespeculators’optimal bid falls within the range of regular bidders’ valuations. Finally, in a Type III equilibrium,noforkoccurs,andthespeculators’optimalbidiszerobecausethe expected redemption value is too low to participate profitably in the auction. We then proceed to explore four different redemption mechanisms and their im- plications for speculators’ incentives. We further consider various extensions of our model, including atomic exits, time delays between exit and redemption, and spending paths of the organization, and provide numerical simulations of our main findings. Ouranalysisaddstotheliteratureonauctionswithspeculatorsbystudyinga settinginwhichspeculatorsbenefitnotfromasecondarymarketbutbydirectly receiving their share of the organization’s assets. Our results also contribute to understanding incentive structures and governance design in auction-based DAOs. The two “innocent” goals of (i) allowing unrestricted participation in governance by bidding for shares (and associated voting rights) and (ii) safe- guarding members from majority attacks by allowing them to exit and redeem their share can combine in practice to yield unintended consequences—such as speculation and costly member exits. Repeated Auctions with Speculators 15 References 1. Bukhchandani,S.,Huang,C.F.:Auctionswithresalemarkets:Anexploratorymodel of treasury bill markets. Rev. Financ. Stud. 2(3), 311–339 (1989). 2. Garratt, R., Tro¨ger, T.: Speculation in Standard Auctions with Resale. Econometrica 74(3), 753–769 (2006). https://ideas.repec.org/a/ecm/emetrp/ v74y2006i3p753-769.html 3. Haile, P.A.: Auctions with resale markets: An application to US Forest Service timber sales. Am. Econ. Rev. 91(3), 399–427 (2001). 4. Lukkarinen,A.,Schwienbacher,A.:Secondarymarketlistingsinequitycrowdfund- ing: The missing link? Res. Policy 52(1), 104648 (2023). 5. Pagnozzi, M.: Are speculators unwelcome in multi-object auctions? Am. Econ. J. Microecon. 2(2), 97–131 (2010). 16 Nicolas Eschenbaum, Nicolas J. Greber A Additional Figures Fig.3: Contribution-based Share 1 and 2. B Tables Table 1: Equilibrium types for different values of the initial treasury (vertical) and forking threshold (horizontal) and Mechanism 1. Repeated Auctions with Speculators 17 (a) Tax of 25%. (b) Tax of 50%. Table 2: Equilibrium types for different values of the initial treasury (vertical) and forking threshold (horizontal) and Mechanism 2. (a) P =0 (b) P =10 Table 3: Equilibrium types for different values of the initial treasury (vertical) and forking threshold (horizontal) and Mechanism 3. 18 Nicolas Eschenbaum, Nicolas J. Greber C Proofs C.1 Proof of Lemma 1: Proof. We prove the statements in turn. i. Let dB denote the density of ˜b . The expected return for nouner i of −i,t −i,t playing a strategy of bidding up to˜b is given by i,t (cid:90) ˜bi,t (v −˜b )dB . (17) i,t −i,t −i,t 0 This expression is maximized at ˜b = v for any strategies of all players i,t i,t other than i that give rise to the density dB . −i,t ii. Fix a Ve(h ) and ˜b (h ). First, the arbitrageur’s utility function directly t t a,t t impliesthatholdingthebid˜b (h )constant,utilityisincreasinginVe(ht), a,t t t implying a maximum bid value ˆb (h ) exists and that ˜b (h ) = ˆb (h ) in t t a,t t t t equilibrium. Second, consider that an arbitrageur’s bid has two effects on herpayoff:changingtheprobabilityofwinningandaffectingtheredemption value by changing today’s auction price and future arbitrageur bids. Thus, if today’s auction price is independent of an arbitrageur’s bid, then so is Ve(h ). This is the case ifˆb (h )>v or ifˆb (h )≤v , where v and t t t t n,t t t n−1,t n,t v arethehighestandsecond-highestvaluationsofnounersathistoryh , n−1,t t respectively. By statement (i) and the definition of the arbitrageurs payoff, it then follows thatˆb≥min{v ,Ve(h )}. ⊔⊓ n−1,t t t C.2 Proof of Lemma 2: Proof. Followsdirectlyfromthedefinitionoftheforkingthresholdκandthefact thatbyLemma1theexpectednumberofarbitrageurwinsisdeterminedbythe sumoverallperiodsτ ∈t,...,t∗ oftherespectiveprobabilitythatv ≤ˆb (h ). n,τ τ τ ⊔⊓ C.3 Proof of Lemma 3: Proof. Consider an arbitrageurs’ payoff, which can be rewritten by Lemma 1 as follows u (˜b ,h )=Prob[˜b >v |h ](Ve(h )−E[v ]) (18a) a,t a,t t a,t n,t t t t n,t (cid:90) ˆba,t = (Ve(h )−v)dB (18b) t t −a,t 0 (cid:90) ˆba,t = (Ve(h )−v)nF(v)n−1f(v)dv, (18c) t t 0 Repeated Auctions with Speculators 19 where v denotes the highest drawn value among all nouners i ∈ 1,...,n at n,t history h and dB the density of the highest nouner valuation v . Differen- t −a,t n,t tiating with respect to the arbitrageurs maximum bidˆb (h ), we obtain a,t t (cid:16) (cid:17) Ve(h )−ˆb nF(ˆb )n−1f(ˆb ) (19a) t t a,t a,t a,t ∂Ve(h )(cid:90) ˆba,t + n t t nF(v)n−1f(v)dv =0 (19b) ∂ˆb a,t 0 (cid:16) (cid:17) ∂Ve(h )1 Ve(h )−ˆb nF(ˆb )n−1f(ˆb )+ t t F(ˆb )n =0 (19c) t t a,t a,t a,t ∂ˆb n a,t a,t Ve(h )−ˆb + ∂V ∂ t ˆb e a ( , h t t)F(ˆb a,t )n =0 (19d) t t a,t nF(ˆb )n−1f(ˆb ) a,t a,t which solves for 1F(b∗)∂Ve(h ) b∗ =Ve(h )+ t t t . (20) t t t n f(b∗) ∂b∗ t t ⊔⊓ C.4 Proof to Proposition 1: Proof. Fix an equilibrium. Let b∗(h ) denote the optimal maximum bid of an t t arbitrageur at history h in equilibrium. Note first that arbitrageur strategies t are symmetric. At any history h and t < τ ≤ t∗ we must therefore have that t E [b∗(h )]=b∗(h )×g(α,τ,h ),whereg(α,τ,h )isaweaklypositive,continuous t τ τ t t t t function. Formally, we can derive g(α,τ,h ) by considering the ratio t b∗ t (h t ) = V t e(h t )+ n 1F f( ( b b ∗ t ∗ t ( ( h h t t ) ) ) )∂ ∂ V b t ∗ t e ( ( h h t t ) ) . E[b∗ τ ] V τ e(h t )+ n 1F f( ( E E [ [ b b ∗ ∗ τ ] ] ) )∂ ∂ V E τ e [ ( b h ∗ t ] ) τ τ We start with the redemption value Ve(h ), or t t   t∗ V t e(h t )=δt∗−tα t∗S t−1 + (cid:88) E[p ϕ ], ϕ=t where E[p ]=Prob[v >b∗]·E[v |v ≥b∗] (21a) ϕ (n−1) ϕ (n−1) (n−1) ϕ +Prob[v ≤b∗ <v ]·b∗ (21b) (n−1) ϕ (n) ϕ +Prob[v ≤b∗]·E[v |v ≤b∗]. (21c) (n) ϕ (n) (n) ϕ The probabilities are given by Prob[v >b∗]=1−n[F(b∗)]n−1[1−F(b∗)]−[F(b∗)]n, (22a) (n−1) ϕ ϕ ϕ ϕ Prob[v ≤b∗ <v ]=n[F(b∗)]n−1[1−F(b∗)], (22b) (n−1) ϕ (n) ϕ ϕ Prob[v ≤b∗]=[F(b∗)]n, (22c) (n) ϕ ϕ 20 Nicolas Eschenbaum, Nicolas J. Greber and the expected values of the second-highest and highest bids are (cid:82)v¯ v·n(n−1)[F(v)]n−2[1−F(v)]f(v)dv E[v |v ≥b∗]= b∗ ϕ , (23a) (n−1) (n−1) ϕ 1−n[F(b∗)]n−1[1−F(b∗)]−[F(b∗)]n ϕ ϕ ϕ (cid:82)bϕv·n[F(v)]n−1f(v)dv E[v |v ≤b∗]= 0 , (23b) (n) (n) ϕ [F(b∗)]n ϕ so that we obtain (cid:18) t∗ (cid:20) Ve(h )=δt∗−tα S + (cid:88) E[b∗]·nF (cid:0)E[b∗] (cid:1)n−1(cid:0) 1−F (cid:0)E[b∗] (cid:1)(cid:1) (24a) t t t t−1 ϕ ϕ ϕ ϕ=t (cid:90) v¯ + v·n(n−1)F(v)n−2(1−F(v))f(v)dv (24b) b∗ ϕ (cid:90) b∗ (cid:21)(cid:19) ϕ + v·nF(v)n−1f(v)dv . (24c) 0 Differentiating the price E[p ] with respect to E[b∗], we find ϕ ϕ ∂E[p ϕ ](cid:0)E[b∗]·n·F(E[b∗])n−1·(1−F(E[b∗])) (cid:1) (25a) ∂E[b∗] ϕ ϕ ϕ ϕ ∂ (cid:90) E[b∗ ϕ ] + v·nF(v)n−1f(v)dv (25b) ∂E[b∗] ϕ 0 ∂ (cid:90) v¯ + v·n(n−1)F(v)n−2(1−F(v))f(v)dv (25c) ∂E[b∗] ϕ E[b∗] ϕ =n·F(E[b∗])n−1·(1−F(E[b∗])) (25d) ϕ ϕ +E[b∗]·n·f(E[b∗]) (cid:0) (n−1)F(E[b∗])n−2(1−F(E[b∗]))−F(b∗)n−1(cid:1) (25e) ϕ ϕ ϕ ϕ t +E[b∗]·nF(E[b∗])n−1f(E[b∗]) (25f) ϕ ϕ ϕ −E[b∗]·n(n−1)F(E[b∗])n−2(1−F(E[b∗]))f(E[b∗]) (25g) ϕ ϕ ϕ ϕ =n·F(E[b∗])n−1·(1−F(E[b∗])) (25h) ϕ ϕ ≥0, (25i) where the final inequality follows from the definition of the CDF F(·), which satisfies0≤F(b∗)≤1.Now,finallysubstitutingintotheratioweareinterested ϕ Repeated Auctions with Speculators 21 in, we obtain b∗ t (ht) (26a) E[b∗] τ = δt∗−tαt (cid:16) St+(cid:80)t ϕ ∗ =t E[pϕ] (cid:17) + n 1F f( ( b b ∗ t ∗ t ) ) δt∗−tαt (cid:80)t ϕ ∗ =t (nF(b∗ ϕ )n−1(1−F(b∗ ϕ ))) (26b) δt∗−ταϕ (St+(cid:80)t τ ∗ =τ E[pτ])+ n 1F f( ( E E [ [ b b ∗ ∗ τ ] ] ) ) δt∗−τατ (cid:80)t τ ∗ =τ (nF(E[b∗ τ ])n−1(1−F(E[b∗ τ ]))) τ = δt∗−t αt (cid:34) (cid:16) St+(cid:80)t τ ∗ =t E[pτ] (cid:17) + n 1F f( ( b b ∗ t ∗ t ) )(cid:80)t τ ∗ =t (nF(b∗ t )n−1(1−F(b∗ t ))) (cid:35) (26c) δt∗−τ ατ (St+(cid:80)t τ ∗ =τ E[pτ])+ n 1F f( ( E E [ [ b b ∗ ∗ τ ] ] ) )(cid:80)t τ ∗ =τ (nF(E[b∗ τ ])n−1(1−F(E[b∗ τ ]))) τ δt∗−t α = t ×a (26d) δt∗−τ α t,τ τ 1 = (26e) g(α,τ,h ) t ≥0 (26f) wherethefinalinequalityfollowsfroma ≥0,wherea isdefinedastheterm t,τ t,τ in square brackets and all of its elements are weakly positive. NowconsiderthenecessaryconditionforaforktooccurinLemma2.Replac- ing the cutoff bids with the increase of expected optimal bids by arbitrageurs, we obtain that the expected time to fork at h is given by t (cid:88) t∗ (cid:20) F (cid:18) b∗· 1 α τ 1 (cid:19)(cid:21)n ≥κ(N +t∗)−A , (27) 1 δτ−t α a t t t,τ τ=t which must be satisfied for a fork to occur in expectation at h and thus at any 1 h . Moreover, as ∂Ve(h )/∂E[p ]≥0, we find that t t t τ ∂V t e(h t ) =δt∗−tα (cid:88) t∗ (cid:18) nF (cid:18) b∗ t α τ 1 (cid:19)n−1 (28a) ∂E[b∗] t δτ−t α a τ τ=t t t,τ (cid:18) (cid:18) b∗ α 1 (cid:19)(cid:19)(cid:19) · 1−F t τ (28b) δτ−t α a t t,τ ≥0. (28c) By Lemma 3 we therefore find that if b∗ ∈ [0,v¯], then b∗ ≥ Ve(h ). In t t t t conjunction,thisyieldstheconditionsforTypeIIandTypeIIIinProposition1 and the sufficient condition for Type II. Lastly, consider the case when b∗ ∈/ (0,v¯]. First, we know by Lemma 3 and 1 the fact that ∂Ve(h )/∂E[p ]>0 that if Ve(h )=v¯, then b∗ ≥v¯. We thus find t t τ t t 1 that if v¯ Ve(h )=δT−tα S ≥v¯ ⇒ S ≥ (29) t t t 0 0 δT−tα t and κ T ≥κ(N +T) ⇒ T ≥ N (30) 1−κ then b∗(h )≥v¯. This completes the proof for Type I. ⊔⊓ t t 22 Nicolas Eschenbaum, Nicolas J. Greber C.5 Proof of Corollary 1: Proof. Consider the redemption value, which under Mechanism 4 and given a bid at time t and history h becomes t (cid:32) t∗ (cid:33) Ve(h )=δt∗−tα (h ) S + (cid:88) E[p ] (31a) t t j,t t 0 τ τ=t (cid:32) t∗ (cid:33) =δt∗−t p j=t S + (cid:88) E[p ] (31b) S + (cid:80)t∗ E[p ] 0 τ 0 τ=t τ τ=t =p δt∗−t. (31c) j=t where without loss of generality we set α (h ) = p /S , as S ≮ 0. An arbi- j,t t j t 0 trageur’s utility, therefore, becomes u (b∗(h ),h )=Prob[b∗(h )>v ](Ve(h )−p ) (32a) a,t t t t t t n,t t t j=t =Prob[b∗(h )>v ](p (δ−1)) (32b) t t n,t j=t <0 ∀ b∗(h )>0, (32c) t t implying that in equilibrium b∗(h )=0. ⊔⊓ t t C.6 Proof of Proposition 2: Proof. Assume κ = 0. Consider the arbitrageur’s expected payoff, which then simplifies to u (b∗,h )= (cid:82)b∗ (V(h )−v)nF(v)n−1f(v)dv, (33a) a,t t 0 t and solves for b∗(h )=V(h )+ 1F(b∗(ht)) ∂V(ht). (34a) t t n f(b∗(ht)) ∂b∗(ht) We know by the proof of Proposition 1 that b∗(h ) ≥ V(h ). We thus find that t t in expectation, the arbitrageur will win the auction if V(h )>v . (35) t n,t Because a fork occurs each time an arbitrageur wins, the treasury will shrink at time t and history h in expectation if t α F(b∗(h ))n(S +E[p ])>E[p ], (36) t t t−1 t t or (cid:18) (cid:19) 1 S > −1 E[p ]. (37) t−1 α F(b∗(h ))n t t t NownotethatasS →0,thisconditionbecomesimpossibletofulfill,implying t−1 that the treasury must increase. Similarly observe that as S →1/α ×v¯, the t−1 t condition is guaranteed to be fulfilled because p ≤ v¯ and F(b∗(h )) → 1 as t t S → 1/α × v¯. Finally, note that we know from the proof of Proposition t−1 t 1 that ∂E[p ]/∂b∗(h ) is a continuous function which, in conjunction with the t t definition of F(v) implies that there is a single solution for a bid that leaves the treasury unchanged in expectation. ⊔⊓ Repeated Auctions with Speculators 23 C.7 Proof of Corollary 2: Proof. We prove the statements in turn. Consider that under Mechanism 3, an arbitrageur has a certain positive profit if he bids b =v¯ if S >P , as t t−1 t−1 E[p ] t (S +E[p ])−E[p ]>0 ∀ S >P (38) P +E[p ] t−1 t t t−1 t−1 t−1 t where E[p ]=E[p |b =v¯]. Under Mechanism 4, in turn, in any equilibrium, the t t t arbitrageur plays b∗(h )=0, as t t (cid:26) E[p ] E[p ] (cid:27) F(b )n min t , t (S +E[p ])−E[p ] (39a) t S +E[p ] P +E[p ] t t t t−1 t t−1 t (cid:26) E[p ] E[p ] (cid:27) ≤min t , t (S +E[p ])−E[p ] (39b) S +E[p ] P +E[p ] t t t t−1 t t−1 t E[p ] ≤ t (S +E[p ])−E[p ] (39c) S +E[p ] t t t t−1 t =0. (39d) where E[p ]=E[p |b ] for any b ∈[0,v¯]. ⊔⊓ t t t t C.8 Proof of Proposition 3: Proof. Consider the forking condition (cid:88) t∗ (cid:20) (cid:18) 1 (cid:19)(cid:21)n F b∗· ≥κ(N +t∗)−A , (40) 1 δτ−t t τ=t optimal bid 1F(b∗(h ))∂Ve(h ) b∗(h )=Ve(h )+ t t t t , (41) t t t t n f(b∗(h )) ∂b∗(h ) t t t t and redemption value under the committed spending path (cid:32) t∗ (cid:33) Ve(h )=δt∗−tα S + (cid:88)(cid:16) E[p ]−kS e−λ(τ−1) (cid:17) . (42) t t t∗ t−1 τ τ−1 τ=t Observe that ∂Ve(h ) ∂b∗(h ) ∂ (cid:80)t∗ (cid:2) F (cid:0) b∗· 1 (cid:1)(cid:3)n t t <0, t t >0, τ=t 1 δτ−t >0, (43) ∂z (S ) ∂Ve(h ) ∂b∗ t t−1 t t 1 which immediately delivers the result. ⊔⊓