Heal Noun OClock

--- title: 'Heal Noun OClock' disqus: hackmd --- Project: Heal Noun O'Clock === ## Overview This technical paper documents the analysis conducted as part of the project for Nouns DAO "Heal Noun O'Clock; Full Spec and Economic Audit of % Exit, An Arbitrage-Free Forking Mechanic". A non-technical summary of the findings will be provided as a companion paper. The analysis has been carried out by [Cryptecon](https://cryptecon.org/home.html). The analysis aims to address the growing issue of arbitrage within the Nouns DAO by modeling and evaluating various exit schemes. The model focuses on the dynamics of the treasury stock, auction prices, and redemption values following a fork and subsequent ragequit by modeling the behaviors of nouners and arbers in the daily auctions. We characterize the possible equilibria of the game and show how the outcome changes based on the initial conditions and treasury management. We also provide numerical simulations of the main results. ## Table of Contents [TOC] # Analysis ## 1. Model Setup Let $S_t$ denote the treasury stock at time $t \in 1,...,T$, with $T>1$, where $S_0 \geq 0$ is the treasury prior to the game commencing. The number of nouns at the start 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 $p_t$ the realized auction price at time $t$. The treasury stock therefore evolves over time according to $S_t = S_0 + \sum_{\tau = 1}^{\tau = t-1}p_\tau$. Nouns can be redeemed for a share $\alpha_t$ of the treasury. We denote the expected value of one noun at time $t$ conditional on the history of the game $h_t$ and for an expected redemption period $t^\ast \geq t$ by $V^e_{t}(h_t)$, where we suppress the subscript of the expected redemption period $t^\ast$ for brevity. We will refer to $V^e_{t}(h_t)$ as the redemption value. In the special case of $t^\ast=t$, we write the redemption value simply as $V^e_{t=t^\ast}(h_t) = V(h_t)$. All players discount future payoffs by the same discount factor $\delta \in (0,1)$. The expected redemption value is thus given by $V^e_{t}(h_t) = \delta^{t^\ast-t} \mathbb{E}[\alpha_{t} S_{t}(h_t)]$, where, as before, we suppress the subscript $t^\ast$ for brevity. We assume that nouners value each noun according to a valuation $\tilde{v} \sim [0, \bar{v}]$, with $\bar{v}>0$ constant over time. Let $F$ denote the distribution function for $\tilde{v}$. We assume that $F$ has a density $f$ that is positive and continuous on $[0, \bar{v}]$ 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., $\tilde{v}$, and their realizations without, e.g., $v$. Arbers (or speculators) in turn value the noun at the expected value of redemption $V^e_t(h_t)$. In other words, we make a sharp distinction between nouners -- who value nouns independent of their redemption value or any possible market value on a secondary market -- and arbers, who only value them for the redemption value. Nouns can only be redeemed, however, after a fork occurs. We assume that only arbers want to fork and when they do will instantly redeem their nouns. For a fork to occur at time $t$, the number of nouns held by arbers, $A_t$, must exceed a constant forking threshold $\kappa \in (0,1)$, given by $\frac{A_t}{N+t} \geq \kappa$. At the start of the game, no arber holds a noun, or $A_0 = 0$. The timing of the model is as follows. At each time $t$, the auction commences with one arber and $n \geq 2$ 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 $A_t \geq \kappa (N+t)$. If it does, then all arbers fork and redeem their nouns and the game ends. 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 arbers leave the DAO and redeem their nouns. The continuation game is therefore simply a repetition of the game with a new starting treasury value $S_0$ and number of nouns $N$. We assume that multiple nouners participate in order to guarantee that the expected auction price is always strictly positive. Formally, the history $h_t$ contains treasury stocks and players' bids for all past periods. However, it is straightforward that the current treasury stock, number of nouns held by arbers at the begining of the period, and sum of past prices contain all information required for players' strategies.[^1] Let the sum of prices be denoted by $P_{t-1} = \sum_{\tau=1}^{t-1} p_\tau$. Then it is sufficient for the sequence of histories to be defined as $h_1 = \{S_0, A_0, P_0\}$, $h_2 = \{S_0, A_0, P_0, S_1, A_1, P_1\}$, etc. The set of histories is denoted by $H_t$. Let $\tilde{b}_{i,t}(\cdot | h_t)$ denote the (possibly randomized) bid at which nouner $i \in 1,...,n$ plans to drop out at history $h_t$ as a function of her realized value $v_{i,t}$ and similarly let $\tilde{b}_{a,t}(\cdot|h_t)$ denote the arbers bid. We further denote by $\tilde{b}_{-i,t}$ the highest of the maximum bids of all players other than $i$ and by $\tilde{b}_{-a,t}$ the highest of the maximum bids of all nouners. For all bids $\tilde{b}_{i,t} \geq 0$ and $v_{i,t} \in [0,\bar{v}]$, nouner i's expected payoff at time $t$ is then given by $$ u_{i,t}(b_{i,t}, v_{i,t}) = Prob \left[ \tilde{b}_{i,t} \geq \tilde{b}_{-i,t} \ | \ h_t \right] \left(v_{i,t}-\tilde{b}_{-i,t} \right). $$ Similarly, for all bids $\tilde{b}_{a,t} \geq 0$, the arber's expected payoff at time $t$ is given by $$ u_{a,t}(b_{a,t}, h_t) = Prob\left[ \tilde{b}_{a,t} > \tilde{b}_{-a,t} \ | \ h_t \right] \left( V^e_{t}(h_t) - \tilde{b}_{-a,t} \right). $$ Ties between nouners are broken at random.[^2] [^1]: Depending on the specification of the share, the sum of past prices need not be part of the history. This is the case for example under the current pro-rata mechanic (see Section 3). [^2]: This assumption is innocuous in equilibrium, as the analysis below shows. ## 2. Preliminaries We begin our analysis by studying the bidding behavior of players at any history $h_t$. __Lemma 1__ (Bidding Strategies)__:__ _In any equilibrium and at any $h_t$,_ _(i) all nouners $i \in 1,...,n$ always bid up to their valuation of the noun, or $\tilde{b}_{i,t} = v_{i,t}$,_ _(ii) all arbers always bid up to a cutoff value $\hat{b}_t(h_t)\geq \min \{ v_{n-1,t},V^e_t(h_t) \}$, or $\tilde{b}_{a,t}(h_t) = \hat{b}_t(h_t) \geq \min \{ v_{n-1,t},V^e_t(h_t) \}$,_ _where $v_{n-1,t}t$ denotes the second-highest value drawn among nounerns $i = 1,...,n$ at history $h_t$._ __Proof:__ We prove the statements in turn. (i) Let $d\mathcal{B}_{-i,t}$ denote the density of $\tilde{b}_{-i,t}$. The expected return for nouner $i$ of playing a strategy of bidding up to $\tilde{b}_{i,t}$ is given by $$ \int_{0}^{\tilde{b}_{i,t}} (v_{i,t}-\tilde{b}_{-i,t})d \mathcal{B}_{-i,t}. $$ This expression is maximized at $\tilde{b}_{i,t} = v_{i,t}$ for any strategies of all players other than $i$ that give rise to the density $d \mathcal{B}_{-i,t}$. (ii) Fix a $V^e_t(h_t)$ and $\tilde{b}_{a,t}(h_t)$. First, the arbers utility function directly implies that holding the bid $\tilde{b}_{a,t}(h_t)$ constant, utility is increasing in $V^e_t(h^t)$, implying a maximum bid value $\hat{b}_t(h_t)$ exists and that $\tilde{b}_{a,t}(h_t) = \hat{b}_t(h_t)$ in equilibrium. Second, consider that an arber's bid has two effects on her payoff: changing the probability of winning and affecting the redemption value by changing todays auction price and thereby future arber bids. Thus, if todays auction price is independent of an arber's bid, then so is $V^e_t(h_t)$. This is the case if $\hat{b}_t(h_t) > v_{n,t}$ or if $\hat{b}_t(h_t) \leq v_{n-1,t}$, where $v_{n,t}$ and $v_{n-1,t}$ are the highest and second-highest valuations of nouners at history $h_t$, respectively. By statement (i) and the definition of the arbers payoff, it then follows that $\hat{b} \geq \min\{v_{n-1,t}, V^e_t(h_t)\}$. $$\tag*{$\blacksquare$}$$ Lemma 1 shows that the equilibrium of the game consists of a pair of simple strategies for nouners and the arber. 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, because an arber's valuation, the redemption value, is a function of her own bid and expected future bids and thus is determined in equilibrium. The second statement then shows that while pinning down arbers' bids in equilibrium is more complex, their optimal strategies still follow a simple structure of bidding up to a well-defined maximum bid $\hat{b}_t(h_t)$ that is at least as high as the lower of the arbers 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 $h_t$ as a function of nouners valuations, since the probability of arbers 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 $h_t$, a necessary condition for $t^\ast$ to exist in expectation is given by_ $$ A_t + \sum_{\tau = t}^{t^\ast} \left( \left[F(\hat{b}_{\tau})\right]^n \right) \geq \kappa (N+t^\ast). $$ __Proof:__ Follows directly from the definition of the forking threshold $\kappa$ and the fact that by Lemma 1 the expected number of arber wins is determined by the sum over all periods $\tau \in t, ..., t^\ast$ of the respective probability that $v_{n,\tau} \leq \hat{b}_{\tau}(h_\tau)$. $$\tag*{$\blacksquare$}$$ Lemma 2 shows that the expected time to fork (and the existence of a forking period at all) is driven by the distribution of nouners' valuations and the expected redemption value of arbers. The higher the redemption value, the more likely it is for the arber to win, since arbers' will always at least outbid nouners whenever possible by Lemma 1. But an arber's win in expectation therefore only depends on the probability of their expected redemption value being above the highest nouner valuation and thus they are simply a function of the distribution function of nouners' valuations. Specifically, the probability of winning conditional on the expected redemption value and resulting optimal maximum bid, $F(\hat{b}_{t})$, must be greater than $\kappa$ for the game to move closer to a fork occuring in expectation. If it does, then in expectation arbers' wins accumulate faster than the 'cost' of a larger number of periods having been played and thus the required number of arber wins increasing (as a function of $\kappa$). But note that Lemma 2 only provides a necessary condition, it does not guarantee that a fork occurs. The critical function of the condition in Lemma 2 and the fact that it only guarantees a fork in expectation is illustrated in Figure 1. Note that the displayed evolution of the expected number of arber wins, $E[A_t]$, is illustrative only. <figure style="text-align: center;"> <img src="https://hackmd.io/_uploads/SyNcSm_SR.png" alt="Figure 1" style="width:100%"/> <figcaption><b>Figure 1: </b>The relationship between the expected number of arber wins and the increasing number of nouns required to reach a fork.</figcaption> </figure> Our final preliminary result now provides an explicit characterization of the optimal bidding behavior of arbers in equilibrium and shows how their behavior is determined by the effect an arber's bid has on the evolution of the redemption value. Note however, that this only applies to interior solutions for the optimal maximum bid, denoted by $b_t^\ast(h_t)$. There can be equilibria in which an arber's maximum bid lies outside the range of nouners valuations. Intuitively, this would for example be the case if the initial treasury is so large that the expected value of a share of it is higher than any nouner is willing to pay for a noun. __Lemma 3__ (Arber Bid) __:__ _In any equilibrium and at any $h_t$, if the optimal maximal bid by an arber satisfies $b_t^\ast(h_t) \in [0,\bar{v}]$, then_ $$ b_t^\ast(h_t) = V^e_t(h_t) + \frac{1}{n} \frac{ F(\hat{b}_{a,t}(h_t))}{f(\hat{b}_{a,t}(h_t))} \frac{\partial V^e_t(h_t)}{\partial \hat{b}_{a,t}(h_t)}. $$ __Proof:__ Consider an arbers' payoff, which can be rewritten by Lemma 1 as follows \begin{align*} u_{a,t}(\tilde{b}_{a,t},h_t) &= Prob[\tilde{b}_{a,t} > v_{n,t} | h_t] (V^e_t(h_t) - E[v_{n,t}]) \\ &= \int_{0}^{\hat{b}_{a,t}} \left( V^e_t(h_t) - v \right) d\mathcal{B}_{-a,t} \\ &= \int_{0}^{\hat{b}_{a,t}} \left( V^e_t(h_t) - v \right) n F(v)^{n-1} f(v) \, dv , \end{align*} where $v_{n,t}$ denotes the highest drawn value among all nouners $i \in 1,...,n$ at history $h_t$ and $d\mathcal{B}_{-a,t}$ the density of the highest nouner valuation $v_{n,t}$. Differentiating with respect to the arbers maximum bid $\hat{b}_{a,t}(h_t)$, we obtain \begin{align*} \left( V^e_t(h_t) - \hat{b}_{a,t} \right) n F(\hat{b}_{a,t})^{n-1} f(\hat{b}_{a,t}) + n \frac{\partial V^e_t(h_t)}{\partial \hat{b}_{a,t}} \int_{0}^{\hat{b}_{a,t}} n F(v)^{n-1} f(v) \, dv &= 0 \\ \left( V^e_t(h_t) - \hat{b}_{a,t} \right) n F(\hat{b}_{a,t})^{n-1} f(\hat{b}_{a,t}) + \frac{\partial V^e_t(h_t)}{\partial \hat{b}_{a,t}} \frac{1}{n}F(\hat{b}_{a,t})^n &= 0 \\ V^e_t(h_t) -\hat{b}_{a,t} + \frac{\frac{\partial V^e_t(h_t)}{\partial \hat{b}_{a,t}} F(\hat{b}_{a,t})^n}{n F(\hat{b}_{a,t})^{n-1} f(\hat{b}_{a,t})} &= 0 \end{align*} which solves for \begin{align*} b^\ast_{t} &= V^e_t(h_t) + \frac{1}{n} \frac{ F(b^\ast_{t})}{f(b^\ast_{t})} \frac{\partial V^e_t(h_t)}{\partial b^\ast_{t}}. \end{align*} $$\tag*{$\blacksquare$}$$ Lemma 3 shows that arbers optimize their bid relative to the redemption value. If the redemption value increases in their bid, then arbers may 'overbid' -- that is, holding the redemption value fix for a given bid, they optimally bid more than this redemption value. If instead the redemption value decreases in their bid, then they would underbid. Whether arbers 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 arbers' strategies implying that while an arbers 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 arbers' strategy. ## 3. Main Results We now provide our main results on the equilibrium outcome of the game. Our first key finding studies the conditions under which a fork occurs in equilibrium. __Proposition 1__ (Equilibrium) _The equilibrium of the game is one of three types which are characterized by the following conditions:_ *__Type I:__* _a fork is guaranteed to occur along the equilibrium path and the optimal maximum bid by arbers satisfies $b_t^\ast(h_t) \geq \bar{v}$ if $T \geq \frac{\kappa}{1-\kappa}N$ and $S_0 \geq \frac{\bar{v}}{\delta^{T-t} \alpha}$,_ *__Type II:__* _a fork is expected to occur along the equilibrium path starting at history $h_t$ and the optimal maximum bid by arbers satisfies $b^\ast_t(h_t) \geq V^e_t(h_t)$ and $b_t^\ast(h_t) \in (0, \bar{v})$ iff $\sum_{\tau=t}^{t^\ast} \left[F \left( b_t^\ast(h_t) \times g(\alpha, \tau, h_t) \right) \right]^n \geq \kappa(N+T) - A_t$ for any $t^\ast \leq T$,_ *__Type III:__* _no fork is expected to occur along the equilibrium path starting at history $h_t$ and the optimal maximum bid by arbers satisfies $b^\ast_t(h_t) = 0$ iff $\sum_{\tau=t}^{t^\ast} \left[F \left( b_t^\ast(h_t) \times g(\alpha, \tau, h_t) \right) \right]^n < \kappa(N+T) - A_t$ for all $t^\ast \leq T$,_ _where $g(\cdot)$ is given by $$ g(\alpha, \tau, h_t) = \frac{1}{\delta^{\tau-t}} \frac{\alpha_{\tau}}{\alpha_t} a_{\tau,t}, $$ and $$ a_{\tau,t} = \left[ \frac{\left( S_t + \sum_{\tau = t}^{t^*} \mathbb{E}[p_\tau] \right) + \frac{1}{n} \frac{F(b_t^*)}{f(b_t^*)} \sum_{\tau = t}^{t^*} \left( n F(b_t^*)^{n-1} (1 - F(b_t^*)) \right)}{ \left( S_t + \sum_{\tau = \tau}^{t^*} \mathbb{E}[p_\tau] \right) + \frac{1}{n} \frac{F(\mathbb{E}[b_{\tau}^*])}{f(\mathbb{E}[b_{\tau}^*])} \sum_{\tau = \tau}^{t^*} \left( n F(\mathbb{E}[b_{\tau}^\ast])^{n-1} (1 - F(\mathbb{E}[b_{\tau}^\ast])) \right)} \right] \geq 0. $$ Furthermore, a sufficient condition for a Type II equilibrium is given by $$ \sum_{\tau=t}^{t^\ast} \left[F \left( V^e_t(h_t) \times g(\alpha,\tau,h_t) \right) \right]^n \geq \kappa(N+T) - A_t \ \textnormal{for any} \ t^\ast \leq T. $$_ __Proof:__ Fix an equilibrium. Let $b_t^\ast (h_t)$ denote the optimal maximum bid of an arber at history $h_t$ in equilibrium. Note first that arber strategies are symmetric. At any history $h_t$ and $t < \tau \leq t^\ast$ we must therefore have that $\mathbb{E}_t[b^\ast_\tau(h_\tau)] = b_t^\ast(h_t) \times g(\alpha,\tau, h_t)$, where $g(\alpha,\tau, h_t)$ is a weakly positive, continuous function. Formally, we can derive $g(\alpha,\tau, h_t)$ by considering the ratio $$ \frac{b_t^\ast (h_t)}{\mathbb{E}[b_{\tau}^\ast]} = \frac{V^e_t(h_t) + \frac{1}{n} \frac{F(b_t^\ast(h_t))}{f(b_t^\ast(h_t))} \frac{\partial V^e_{t}(h_t)}{\partial b_t^\ast(h_t)}} {V^e_{\tau}(h_t) + \frac{1}{n} \frac{F(\mathbb{E}[b_{\tau}^\ast])}{f(\mathbb{E}[b_{\tau}^\ast])} \frac{\partial V^e_{\tau}(h_t)}{\partial \mathbb{E}[b_{\tau}^\ast]}}. $$ We start with the redemption value $V^e_t(h_t)$, or $$ V^e_t(h_t) = \delta^{t^\ast-t} \alpha_{t^\ast} \left( S_{t-1} + \sum_{\phi = t}^{t^\ast} \mathbb{E}[p_\phi] \right), $$ where \begin{align*} \mathbb{E}[p_{\phi}] &= Prob[v_{(n-1)} > b_\phi^\ast] \cdot \mathbb{E}[v_{(n-1)} \vert v_{(n-1)}\geq b_\phi^\ast]\\ & + Prob[v_{(n-1)} \leq b_\phi^\ast < v_{(n)}] \cdot b_\phi^\ast\\ &+ Prob[v_{(n)} \leq b_\phi^\ast] \cdot \mathbb{E}[v_{(n)} \vert v_{(n)}\leq b_\phi^\ast]. \end{align*} The probabilities are given by \begin{align*} Prob[v_{(n-1)} > b_\phi^\ast] &= 1 - n[F(b_\phi^\ast)]^{n-1}[1 - F(b_\phi^\ast)] - [F(b_\phi^\ast)]^n,\\ Prob[v_{(n-1)} \leq b_\phi^\ast < v_{(n)}] &= n[F(b_\phi^\ast)]^{n-1}[1 - F(b_\phi^\ast)], \\ Prob[v_{(n)} \leq b_\phi^\ast] &= [F(b_\phi^\ast)]^n, \end{align*} and the expected values of the second-highest and highest bids are \begin{align*} \mathbb{E}[v_{(n-1)} \vert v_{(n-1)}\geq b_\phi^\ast] &= \frac{\int_{b_{\phi}^\ast}^{\bar{v}} v \cdot n (n-1) [F(v)]^{n-2} [1 - F(v)] f(v) \, dv}{1 - n[F(b_\phi^\ast)]^{n-1}[1 - F(b_\phi^\ast)] - [F(b_\phi^\ast)]^n}, \\ \mathbb{E}[v_{(n)} \vert v_{(n)}\leq b_\phi^\ast] &= \frac{\int_0^{b_\phi} v \cdot n [F(v)]^{n-1} f(v) \, dv}{[F(b_\phi^\ast)]^n}, \end{align*} so that we obtain \begin{align*} V^e_t(h_t) &= \delta^{t^\ast-t} \alpha_{t} \left( S_{t-1} + \sum_{\phi=t}^{t^\ast} \biggl[ \mathbb{E}[b_\phi^\ast] \cdot nF\left(\mathbb{E}[b_\phi^\ast] \right)^{n-1}\left(1 - F\left(\mathbb{E}[b_\phi^\ast] \right)\right) \biggr. \right. \\ &\quad \left. \biggl. + \int_{b_\phi^\ast}^{\bar{v}} v \cdot n(n-1)F(v)^{n-2}(1 - F(v)) f(v) \, dv + \int_0^{b_\phi^\ast} v \cdot nF(v)^{n-1} f(v) \, dv \biggr] \right) \end{align*} Differentiating the price $\mathbb{E}[p_\phi]$ with respect to $\mathbb{E}[b_\phi^\ast]$, we find \begin{align*} \frac{\partial \mathbb{E}[p_\phi]}{\partial \mathbb{E}[b_\phi^\ast]} & \left( \mathbb{E}[b_\phi^\ast] \cdot n \cdot F(\mathbb{E}[b_\phi^\ast])^{n-1} \cdot (1 - F(\mathbb{E}[b_\phi^\ast])) \right) + \frac{\partial}{\partial \mathbb{E}[b_\phi^\ast]} \int_0^{\mathbb{E}[b_\phi^\ast]} v \cdot nF(v)^{n-1} f(v) \, dv \\ &\quad + \frac{\partial}{\partial \mathbb{E}[b_\phi^\ast]} \int_{\mathbb{E}[b_\phi^\ast]}^{\bar{v}} v \cdot n(n-1)F(v)^{n-2}(1 - F(v)) f(v) \, dv \\[1em] &\quad = n \cdot F(\mathbb{E}[b_\phi^\ast])^{n-1} \cdot (1 - F(\mathbb{E}[b_\phi^\ast])) \\ &\quad \quad + \mathbb{E}[b_\phi^\ast] \cdot n \cdot f(\mathbb{E}[b_\phi^\ast]) \left( (n-1) F(\mathbb{E}[b_\phi^\ast])^{n-2} (1 - F(\mathbb{E}[b_\phi^\ast])) - F(b_t^\ast)^{n-1} \right)\\ &\quad \quad + \mathbb{E}[b_\phi^\ast] \cdot nF(\mathbb{E}[b_\phi^\ast])^{n-1} f(\mathbb{E}[b_\phi^\ast])\\ &\quad \quad - \mathbb{E}[b_\phi^\ast] \cdot n(n-1)F(\mathbb{E}[b_\phi^\ast])^{n-2}(1 - F(\mathbb{E}[b_\phi^\ast])) f(\mathbb{E}[b_\phi^\ast]) \\[1em] &\quad = n \cdot F(\mathbb{E}[b_\phi^\ast])^{n-1} \cdot (1 - F(\mathbb{E}[b_\phi^\ast])) \\[1em] &\quad \geq 0, \end{align*} where the final inequality follows from the definition of the CDF $F(\cdot)$, which satisfies $0 \leq F(b_\phi^\ast) \leq 1$. Now finally substituting into the ratio we are interested in, we obtain \begin{align*} \frac{b_t^\ast (h_t)}{\mathbb{E}[b_{\tau}^\ast]} &= \frac{\delta^{t^* - t} \alpha_t \left( S_t + \sum_{\phi = t}^{t^*} \mathbb{E}[p_\phi] \right) + \frac{1}{n} \frac{F(b_t^*)}{f(b_t^*)} \delta^{t^* - t} \alpha_t \sum_{\phi = t}^{t^*} \left( n F(b_\phi^*)^{n-1} (1 - F(b_\phi^*)) \right)}{\delta^{t^* - \tau} \alpha_{\phi} \left( S_t + \sum_{\tau = \tau}^{t^*} \mathbb{E}[p_\tau] \right) + \frac{1}{n} \frac{F(\mathbb{E}[b_{\tau}^*])}{f(\mathbb{E}[b_{\tau}^*])} \delta^{t^* - \tau} \alpha_{\tau} \sum_{\tau = \tau}^{t^*} \left( n F(\mathbb{E}[b_{\tau}^\ast])^{n-1} (1 - F(\mathbb{E}[b_{\tau}^\ast])) \right)} \\[1em] &= \frac{\delta^{t^\ast-t}}{\delta^{t^\ast-\tau}} \frac{\alpha_t}{\alpha_\tau} \left[ \frac{\left( S_t + \sum_{\tau = t}^{t^*} \mathbb{E}[p_\tau] \right) + \frac{1}{n} \frac{F(b_t^*)}{f(b_t^*)} \sum_{\tau = t}^{t^*} \left( n F(b_t^*)^{n-1} (1 - F(b_t^*)) \right)}{ \left( S_t + \sum_{\tau = \tau}^{t^*} \mathbb{E}[p_\tau] \right) + \frac{1}{n} \frac{F(\mathbb{E}[b_{\tau}^*])}{f(\mathbb{E}[b_{\tau}^*])} \sum_{\tau = \tau}^{t^*} \left( n F(\mathbb{E}[b_{\tau}^\ast])^{n-1} (1 - F(\mathbb{E}[b_{\tau}^\ast])) \right)} \right] \\ &= \frac{\delta^{t^\ast-t}}{\delta^{t^\ast-\tau}} \frac{\alpha_t}{\alpha_\tau} \times a_{t,\tau} \\ &= \frac{1}{g(\alpha, \tau, h_t)} \\[1em] &\geq 0 \end{align*} where the final inequality follows from $a_{t,\tau} \geq 0$, where $a_{t,\tau}$ is defined as the term in square brackets and all of its elements are weakly positive. Now consider the necessary condition for a fork to occur in Lemma 2. Replacing the cutoff bids with the increase of expected optimal bids by arbers, we obtain that the expected time to fork at $h_t$ is given by $$ \sum_{\tau = t}^{t^\ast} \left[F\left(b_1^\ast \cdot \frac{1}{\delta^{\tau-t}}\frac{\alpha_\tau}{\alpha_t} \frac{1}{a_{t,\tau}}\right)\right]^n \geq \kappa (N + t^\ast) - A_t, $$ which must be satisfied for a fork to occur in expectation at $h_1$ and thus at any $h_t$. Moreover, as $\partial V_t^e(h_t) / \partial \mathbb{E}[p_\tau] \geq 0$, we find that $$ \frac{\partial V_t^e(h_t)}{\partial \mathbb{E}[b_\tau^\ast]} = \delta^{t^\ast-t} \alpha_{t} \sum_{\tau=t}^{t^\ast} \left( n F\left(\frac{b_t^\ast}{\delta^{\tau-t}} \frac{\alpha_\tau}{\alpha_t} \frac{1}{a_{t,\tau}} \right)^{n-1} \cdot \left(1 - F\left(\frac{b_t^\ast}{\delta^{\tau-t}} \frac{\alpha_\tau}{\alpha_t} \frac{1}{a_{t,\tau}} \right) \right) \right) \geq 0. $$ By Lemma 3 we therefore find that if $b_t^\ast \in [0,\bar{v}]$, then $b_t^\ast \geq V^e_t(h_t)$. In conjunction, this yields the conditions for Type II and Type III in Proposition 1 and the sufficient condition for Type II. Lastly, consider the case when $b_1^\ast \notin (0,\bar{v}]$. First, we know by Lemma 3 and the fact that $\partial V^e_t(h_t)/\partial \mathbb{E}[p_\tau] > 0$ that if $V^e_t(h_t) = \bar{v}$, then $b_1^\ast \geq \bar{v}$. We thus find that if $$ V^e_t(h_t) = \delta^{T-t} \alpha_t S_0 \geq \bar{v} \ \implies \ S_0 \geq \frac{\bar{v}}{\delta^{T-t} \alpha_t} $$ and $$ T \geq \kappa (N+T) \ \implies \ T \geq \frac{\kappa}{1-\kappa}N $$ then $b_t^\ast(h_t) \geq \bar{v}$. This completes the proof for Type I. $$\tag*{$\blacksquare$}$$ Proposition 1 characterizes the three possible types of equilibria and provides conditions under which the game will end in each type of equilibrium. Intuitively, the result of the game depends on whether arbers' optimal bids lie within the interval of the distribution of nouner valuations. If they do and are sufficiently high, then the accumulating expected wins by arbers can be high enough for arbers to expect a fork to occur and be willing to bid a positive amount (Type II). If they are not sufficiently high, then in expectation no fork will occur before the game ends, so that arbers are not willing to bid any positive amount (Type III). Finally, if they lie outside the bounds of the distribution, then arbers will always win and as long as the game lasts long enough for them 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 $S_0$ is so large that a Type I equilibrium arises. There can be intermediate parameter values that also result in Type I equilibria being played 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 arbers to result in the respective type of equilibrium, but rather are expressed in terms of the optimal arber 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. Spelling out the condition in more detail thus does not provide any additional precision or intuition. However, the conditions show that in order to characterize the equilibrium outcome of the model, it is sufficient to check whether at the beginning of the game arbers 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 arbers, 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 arbers (shaded area) represent the expected number of arber wins and if this is sufficiently high, then a fork is expected to occur. In this case the positive bids by arbers 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 arber bid of zero and corresponding probability of winning of zero, while in a Type I equilibrium, the optimal bid lies above the bound of nouners valuations $\bar{v}$ (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 continuous functions while in the model time is discrete. <figure style="text-align: center;"> <img src="https://hackmd.io/_uploads/r1EXlGI4R.png" alt="Figure 2" style="width:100%"/> <figcaption><b>Figure 2: </b>Example of expected optimal future bids at period 1 and resulting expected accumulating arber wins over time for the case of nouners valuations being distributed according to a normal distribution.</figcaption> </figure> Lastly, consider that the conditions stated in Proposition 1 hold for various possible assumptions relevant for the redemption value. In particular, we have not specified the share arbers receive when redeeming a noun. The conditions stated in Proposition 1 only assume that the share arbers receive is not decreasing in their bid, that is, $\partial \alpha_{t^\ast}/\partial b_t^\ast(h_t) \geq 0$. We now proceed to numerically solve the model under different assumptions on the share arbers receive when redeeming a noun. We begin by imposing the current mechanism in the DAO on the model, specifically: __Mechanism 1__ (Pro-rata share): $\alpha_t = \frac{1}{N+t}$. That is, when redeeming a noun, players receive the pro rata share of the entire treasury. Table 1 shows for the case of Mechanism 1 how different values for the forking threshold $\kappa$ and the initial treasury $S_0$ affect the outcome of the game for one particular set of parameter values and distribution function of nouners. As expected, lower forking thresholds or higher initial treasuries reduce 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 initial treasury of zero, a fork can still occur. This is because it can be profitable for arbers to play relative low, but positive maximum bids, resulting in nouners winning in expectation for a number rounds and paying their prices into the treasury. Once arbers are able to force a fork, the nouners' paid prices result in a net gain for arbers. For this to be an equilibrium strategy, however, the forking threshold ($\kappa$) must be quite low. <figure style="text-align: center;"> <img src="https://hackmd.io/_uploads/r1zUxjzDC.png" alt="Table 1" style="width:70%"/> <figcaption><b>Table 1: </b>Equilibrium types for different values of the initial treasury (vertical) and forking threshold (horizontal) and Mechanism 1.</figcaption> </figure> The simulation runs for Table 1 were conducted with the following parameters: the maximum number of periods was set to $T = 30$, with number of initial nouners $N = 10$ and a discount factor $\delta = 0.95$. The number of nouners participating in the auctions was set at $n = 2$ and the upper bound of the uniform distribution at $\bar{v} = 1$. The second assumption on the share arbers receive when redeeming the noun that we study is the following: __Mechanism 2__ (Pro-rata share with tax): $\alpha_t = c \times \frac{1}{N+t}$ with $c\in (0,1)$. In this case, arbers only receive a fraction of their pro-rata share. The remaining sum either stays in the DAOs treasury or is burned. It is straightforward 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 remaining sum stays in the DAOs treasury, then our analysis is exactly identical to before. 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. Table 2 and Table 3 document the numerical results for Mechanism 2. The effect of setting a tax is similar to the effect of a higher $\kappa$ or lower initial treasury: it reduces the redemption value and thereby the willingness to pay by arbers. In our simulations, the tax parameter ($c$) was set to $0.75$ and $0.5$, equivalent to a 25% and 50% tax, respectively. <figure style="text-align: center;"> <img src="https://hackmd.io/_uploads/SkHJWifvA.png" alt="Table 2" style="width:70%"/> <figcaption><b>Table 2: </b>Equilibrium types for different values of the initial treasury (vertical) and forking threshold (horizontal) and Mechanism 2 with a tax of 25%. </figcaption> </figure> <figure style="text-align: center;"> <img src="https://hackmd.io/_uploads/Hko3-szv0.png" alt="Table 3" style="width:70%"/> <figcaption><b>Table 3: </b>Equilibrium types for different values of the initial treasury (vertical) and forking threshold (horizontal) and Mechanism 2 with a tax of 50%. </figcaption> </figure> As the tax rate increases, equilibrium conditions are met at lower initial treasury levels or higher forking thresholds. Lowering $c$ makes forking less likely and 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, whereas changing the initial treasury size more directly impacts the profitability of bidding for arbers. :::info **Taxed Share** Mechanism 2 raises the question of what happens to the taxed share of arbers' redemption value. There are three possibilities, each of which can be analyzed using our model and results. - *The funds remain with the old DAO:* In this case, the continuation game of our model starts with a larger initial treasury relative to when the entire funds are burnt or transferred to the new DAO/the arbers. All our results then apply again and thus we see that because of the greater initial treasury, the next fork becomes more likely and occurs earlier in expectation than otherwise. - *The funds remain with the new DAO:* In this case, the continuation game for the old DAO is unaffected, but the new DAO now has a positive initial treasury and since it allows for a ragequit, its forking mechanism is equivalent to the atomic exit mechanism we analyze in section 3.1. However, arbers may not be able to obtain NFTs for the new DAO via the same auction mechanism as for Nouns DAO, so while the intuition of the results from 3.1 carries over, the details may differ. - *The funds are burned:* In this case, the continuation game of the old DAO is unaffected again and the tax simply has the effect we document above of lowering the incentives to participate and to fork by arbers. One further aspect to consider is that taxing the redemption value will also have an effect on nouners who may have genuine, non-financial reasons to fork and may judge the tax to be unfair. In the context of our model, this could lead to nouners' valuations of the DAO to decrease or to no longer participate in the daily auctions, making arber wins more likely. ::: The third forking mechanism that we consider is the following, where we denote by $j \leq t$ the time period in which a noun was bought: __Mechanism 3__ (Contribution-based share 1): $\alpha_{j,t}(h_t) = \frac{p_j}{\sum_{\tau=1}^{t} p_{\tau}}$. 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 function of the price paid for the noun or the contribution the owner of the noun has made to the size of the treasury.  Table 4 and Table 5 document the simulation results for Mechanism 3. Compared to the pro-rata mechanism, Mechanism 3 leads to more "extreme" equilibria, that is, Type II equilibria become very unlikely and in almost all scenarios, arbers 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, arbers 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 $t^\ast$. Thus, due to discounting they will value this return slightly less than the price they paid and Type II equilibria become possible again. But this can only be the case under very specific parameter combinations, making Type II equilibria very rare. <figure style="text-align: center;"> <img src="https://hackmd.io/_uploads/BkWlkTfDR.png" alt="Table 4" style="width:70%"/> <figcaption><b>Table 4: </b>Equilibrium types for different values of the initial treasury (vertical) and forking threshold (horizontal) and Mechanism 3 and P=0. </figcaption> </figure> <figure style="text-align: center;"> <img src="https://hackmd.io/_uploads/S1t_OafvC.png" alt="Table 5" style="width:70%"/> <figcaption><b>Table 5: </b>Equilibrium types for different values of the initial treasury (vertical) and forking threshold (horizontal) and Mechanism 3 and P=10. </figcaption> </figure> :::info **Secondary Market** Because nouns are tradeable NFTs, they may be bought and sold on a secondary market outside of the auction mechanism that we consider in the model. Incorporating the behaviour of players on a secondary market is outside the scope of the analysis, but the main effects of a secondary market are the following. First, arbers may buy nouns on the secondary market instead of in the daily auction. Whenever the prices for nouns on the secondary market are lower than the redemption value, buying nouns to force a fork becomes profitable. Moreover, if the secondary market is sufficiently liquid, forks become much more frequent, because potentially $\kappa (N+T)$ nouns can be bought in a single period and arbers can fork immediately. Their redemption value thus is not discounted, because they do not need to wait (or need to wait less) and hence the incentives to fork are stronger. Second, arbers may sell nouns on the secondary market. The price arbers can obtain for selling the noun sets a lower bound on their maximum bid in the auction, because even if they dont expect a fork to occur, they can always sell the noun on the secondary market. This increases the probability of arbers winning in the auction, increasing the probability of a fork occuring and lowering the time to fork in expectation. Overall, a secondary market makes forking (much) more likely. But note that in this analysis, we assume that nouners' incentives in the auction are unaffected by the existence of a secondary market. Finally, the interrelation of the prices on the two markets (the auction and the secondary market) is complex, as seen in the effect on arbers' bids limit above. For example, the ability to win a noun in the auction will similarly limit prices on the secondary market, while the redemption value will determine the willingness to pay for arbers on the secondary market. The exact determination of market prices is complex and depends on the specification of the market mechanism. ::: The final mechanism that we consider is the following: __Mechanism 4__ (Contribution-based share 2): $\alpha_{j,t}(h_t) = \min \left\{ \frac{p_j}{S_{t}}, \frac{p_j}{\sum_{\tau=1}^{t} p_{\tau}} \right\}$. Compared to Mechanism 3, Mechanism 4 puts a cap on the share arbers may receive. Just like in Mechanism 3, arbers receive a share proportional to the auction price they paid relative to sum of auction prices, $p_j/P_t$. But 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 4 illustrates the logic of this mechanism. The redemption value for arbers does not continuously grow with the size of the treasury, instead it is capped at the level of their paid auction price relative to the overall size of the treasury. <figure style="text-align: center;"> <img src="https://hackmd.io/_uploads/S1bejX_rC.png" alt="Figure 4" style="width:50%"/> <figcaption><b>Figure 4: </b>Contribution-based Share 1 and 2.</figcaption> </figure> 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:__ Consider the redemption value, which under Mechanism 4 and given a bid at time $t$ and history $h_t$ becomes \begin{align*} V^e_t(h_t) &= \delta^{t^\ast -t} \alpha_{j,t}(h_t) \left( S_0 + \sum_{\tau=t}^{t^\ast} \mathbb{E}\left[p_\tau \right] \right) \\ &= \delta^{t^\ast - t} \frac{p_{j=t}}{S_0 + \sum_{\tau=t}^{t^\ast} \mathbb{E}\left[p_\tau \right]} \left( S_0 + \sum_{\tau=t}^{t^\ast} \mathbb{E}\left[p_\tau \right] \right) \\ &=p_{j=t} \delta^{t^\ast - t}. \end{align*} where without loss of generality we set $\alpha_{j,t}(h_t) = p_j/S_t$, as $S_0 \nless 0$. An arbers utility therefore becomes \begin{align*} u_{a,t}(b^\ast_t(h_t),h_t) &= Prob[b^\ast_t(h_t) > v_{n,t}] (V^e_t(h_t) - p_{j=t}) \\ &=Prob[b^\ast_t(h_t) > v_{n,t}] (p_{j=t}(\delta - 1) ) \\ &< 0 \ \forall \ b_t^\ast(h_t) > 0, \end{align*} implying that in equilibrium $b_t^\ast(h_t) = 0$. $$\tag*{$\blacksquare$}$$ <figure style="text-align: center;"> <img src="https://hackmd.io/_uploads/S1BsKaGDR.png" alt="Table 6" style="width:70%"/> <figcaption><b>Table 6: </b>Equilibrium types for different values of the initial treasury (vertical) and forking threshold (horizontal) and Mechanism 4. </figcaption> </figure> The simulation results in Table 6 show that, as proved formally in Corollary 1, for any combination of $\kappa$ and $S_0$, the equilibrium is always of Typ III, i.e. "no fork". # 3. Extensions ## 3.1 Atomic Exit Instead of players only being able to fork once a sufficient number of nouns/players all choose to fork at the same time, players can be allowed to exit individually at any point in time (an 'atomic exit'). This mechanism can be captured in our model by placing two assumptions on the framework. First, we assume that $\kappa = 0$. As a consequence, the forking condition in Lemma 2 is no longer required, as it will always be fulfilled once an arber wins their auction. Second, the redemption value for arbers simplifies to $$ V^e_t(h_t) = V(h_t) = \alpha_t (S_{t-1} + \mathbb{E}[p_t]), $$ as the arber can fork after winning at time $t$ immediately and thus her value for the noun no longer depends on future auction prices or expected future arber wins. We can then state the following result, where we also characterize 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 $\kappa=0$ so that players may fork and redeem their noun at any time $t$. Then, in any equilibrium and at any history $h_t$, a fork (atomic exit) occurs in expectation if_ $$ b^\ast(h_t) \in [0, \bar{v}] \ \ \text{and} \ \ F\left( V(h_t) \right) > F(\mathbb{E}[v_{n,t}]) , $$ _and the expected value of the treasury follows a mean-reverting process denoted by $\tilde{S_t}$ and defined by_ $$ \tilde{S_t} \sim \mathcal{D}(\mu_t, \underline{S}, \bar{S}), $$ _where_ $$ \mu_t = (1 - \alpha_t F(b^\ast(h_t))^n)(\mu_t + \mathbb{E}[p_t]), \quad \underline{S} > 0, \quad \bar{S} < \frac{1}{\alpha_t} \bar{v}. $$ __Proof:__ Assume $\kappa = 0$. Consider the arbers expected payoff, which then simplifies to \begin{align*} u_{a,t}(b^\ast,h_t) &= \int_{0}^{b^\ast} \left( V(h_t) - v \right) n F(v)^{n-1} f(v) \, dv , \end{align*} and solves for \begin{align*} b^\ast(h_t) &= V(h_t) + \frac{1}{n} \frac{ F(b^\ast(h_t))}{f(b^\ast(h_t))} \frac{\partial V(h_t)}{\partial b^\ast(h_t)}. \end{align*} We know by the proof of Proposition 1 that $$ \frac{\partial V(h_t)}{\partial b^\ast(h_t)} \geq 0 $$ implying that $b^\ast(h_t) \geq V(h_t)$. We thus find that in expectation the arber will win the auction if $$ V(h_t) > v_{n,t}. $$ Because a fork occurs each time an arber wins, the treasury will shrink at time $t$ and history $h_t$ in expectation if $$ \alpha_t F(b^\ast(h_t))^n (S_{t-1} + \mathbb{E}[p_t]) > \mathbb{E}[p_t], $$ or $$ S_{t-1} >\left( \frac{1}{\alpha_t F(b^\ast(h_t))^n} -1 \right) \mathbb{E}[p_t]. $$ Now note that as $S_{t-1} \rightarrow 0$, this condition becomes impossible to fullfill, implying that the treasury must increase. Similarly observe that as $S_{t-1} \rightarrow 1/\alpha_t \times \bar{v}$, the condition is guaranteed to be fullfilled because $p_t \leq \bar{v}$ and $F(b^\ast(h_t)) \rightarrow 1$ as $S_{t-1} \rightarrow 1/\alpha_t \times \bar{v}$. Finally note that we know from the proof of Proposition 1 that $\partial \mathbb{E}[p_t]/\partial b^\ast(h_t)$ is a continuous function which in conjunction with the definition of $F(v)$ implies that there is a single solution for a bid that leaves the treasury unchanged in expectation. $$\tag*{$\blacksquare$}$$ As Proposition 2 shows, in a steady state, the treasury will follow a treasury path defined by the exit mechanism and the expected value of the second highest nouner. Any treasury values that exceed this path will immediately result (in expectation) in an arber buying a noun to exit with profit. The consequence of this change to the forking mechanic in the DAO is particularly stark under Mechanism 3 and 4, as the following statement shows. **Corollary 2:** (Arbitrage under atomic exit)**:** _At any time $t$ and history $h_t$, in any equilibrium:_ - _under Mechanism 3, a fork occurs with certainty whenever $S_{t-1} > P_{t-1}$._ - _under Mechanism 4, no fork occurs._ **Proof:** We prove the statements in turn. Consider that under Mechanism 3, an arber has a certain positive profit if he bids $b_t = \bar{v}$ if $S_{t-1}>P_{t-1}$, as $$ \frac{\mathbb{E}[p_t]}{P_{t-1} +\mathbb{E}[p_t]} (S_{t-1} + \mathbb{E}[p_t]) - \mathbb{E}[p_t] > 0 \quad \forall \ S_{t-1}>P_{t-1} $$ where $\mathbb{E}[p_t] = \mathbb{E}[p_t \vert b_t = \bar{v}]$. Under Mechanism 4 in turn, in any equilibrium the arber plays $b_t^\ast(h_t) = 0$, as \begin{align*} & F(b_t)^n \min \left\{ \frac{\mathbb{E}[p_t]}{S_{t-1} + \mathbb{E}[p_t]}, \frac{\mathbb{E}[p_t]}{P_{t-1} + \mathbb{E}[p_t]} \right\} (S_t + \mathbb{E}[p_t]) - \mathbb{E}[p_t] \\ & \leq \min \left\{ \frac{\mathbb{E}[p_t]}{S_{t-1} + \mathbb{E}[p_t]}, \frac{\mathbb{E}[p_t]}{P_{t-1} + \mathbb{E}[p_t]} \right\} (S_t + \mathbb{E}[p_t]) - \mathbb{E}[p_t] \\ & \leq \frac{\mathbb{E}[p_t]}{S_{t-1} + \mathbb{E}[p_t]} (S_t + \mathbb{E}[p_t]) - \mathbb{E}[p_t] \\ & = 0. \end{align*} where $\mathbb{E}[p_t] = \mathbb{E}[p_t \vert b_t]$ for any $b_t \in [0,\bar{v}]$.$$\tag*{$\blacksquare$}$$ With the atomic exit mechanic, arbers no longer need to consider future expected arber 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 under Mechanism 3 and a positive initial treasury. Under Mechanism 4, however, the arbers' returns are capped and can never be strictly positive and thus we obtain the finding from Corollary 1 again that arbers will refrain from participating in the auction altogether. ## 3.2 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 influences the willingness of arbers to participate. As the delay for receiving funds increases, the willingness of arbitrageurs to participate decreases. This vesting period can be represented with a parameter $\Delta \in 1,2,...,T-1$ that represents the number of time periods (i.e., days) during which the funds are locked. The redemption value then becomes: $$ V^e_t(h_t) = \delta^{ t^\ast-t + \Delta} \alpha_{t^\ast} \left( S_{t-1} + \sum_{\tau = t}^{t^\ast} \mathbb{E}[p_\tau] \right). $$ The main consequence of this change to the forking mechanism is that it reduces the immediate financial incentive for arbers 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 1__ (Vesting)__:__ _Assume funds received from redeeming a noun are vested for $\Delta$ 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 optimal maximum bid by arbers satisfies $b_t^\ast(h_t) \geq \bar{v}$ if $T \geq \frac{\kappa}{1-\kappa}N$ and $S_0 \geq \frac{\bar{v}}{\delta^{T-t+\Delta} \alpha}$._ 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 formal definition does not change but the set of parameters for which they apply changes. Similarly our condition for a Type I equilibrium now requires a larger initial treasury to guarantee a fork. :::info __Vesting__ One additional significant downside to this approach is that participants are exposed to fluctuations in the value of ETH during the vesting period, adding a layer of financial uncertainty that may disincentivize nouners to ragequit from majority attacks. This exposure to ETH price volatility can affect the overall value received by Nouners, potentially deterring participation from those who are risk-averse or require more immediate liquidity. ::: Tables 7 to 12 show that time delay, as expected, reduces the likelihood of a fork occurring and hence increases the number of Type III equilibria. As discussed before, under Mechanism 3 Type II equilibria are once again very unlikely to occur and arbers tend to either win with certainty (forking a fork with certainty), or refrain from participating in the auction. <figure style="text-align: center; display: flex; justify-content: space-around;"> <div style="text-align: center;"> <img src="https://hackmd.io/_uploads/r1zUxjzDC.png" alt="Table 4" style="width:95%"/> <figcaption style="font-size: small;"> 7: Mechanism 1 without vesting </figcaption> </div> <div style="text-align: center;"> <img src="https://hackmd.io/_uploads/SJo5h6GvA.png" alt="Table 4" style="width:95%"/> <figcaption style="font-size: small;"> 8: Mechanism 1 with 10 periods vesting.</figcaption> </div> </figure> <figcaption style="text-align: center;"><b>Tables 7-8:</b> Equilibrium types for different values of the initial treasury (vertical) and forking threshold (horizontal) and Mechanism 1 and vesting of 0 (left) and 10 periods (right).</figcaption> <figure style="text-align: center; display: flex; justify-content: space-around;"> <div style="text-align: center;"> <img src="https://hackmd.io/_uploads/BkWlkTfDR.png" alt="Table 4" style="width:95%"/> <figcaption style="font-size: small;"> 9: Mechanism 3 with P=0 and without vesting </figcaption> </div> <div style="text-align: center;"> <img src="https://hackmd.io/_uploads/BkU560fDA.png" alt="Table 4" style="width:95%"/> <figcaption style="font-size: small;"> 10: Mechanism 3 with P=0 and 10 periods vesting.</figcaption> </div> </figure> <figure style="text-align: center; display: flex; justify-content: space-around;"> <div style="text-align: center;"> <img src="https://hackmd.io/_uploads/S1t_OafvC.png" alt="Table 4" style="width:95%"/> <figcaption style="font-size: small;"> 11: Mechanism 3 with P=10 and without vesting </figcaption> </div> <div style="text-align: center;"> <img src="https://hackmd.io/_uploads/HkROw1XPR.png" alt="Table 4" style="width:95%"/> <figcaption style="font-size: small;"> 12: Mechanism 3 with P=10 and 10 periods vesting.</figcaption> </div> </figure> <figcaption style="text-align: center;"><b>Tables 9-12:</b> Equilibrium types for different values of the initial treasury (vertical) and forking threshold (horizontal) and Mechanism 3 and vesting of 0 (left) and 10 periods (right), P=0 (upper) and P=10 (lower).</figcaption> ## 3.3 Treasury Spending In our baseline model, all outflows from the treasury arise from players forking 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 arbers 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. Let $z_t(h_t)$ denote treasury spending at time $t$ and history $h_t$. We will assume that the level of treasury spending is only a function of the current treasury stock at the beginning of the period, so that $$ z_t(h_t) = z_t(S_{t-1}). $$ The treasury stock therefore evolves over time according to $$ S_t = S_{t-1} + p_t - z_t(S_{t-1}). $$ 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 arbers now needs to be adjusted for the expected level of treasury spending, or $$ V^e_t(h_t) = \delta^{t^\ast-t} \alpha_{t^\ast} \left( S_{t-1} + \sum_{\tau = t}^{t^\ast} \left( \mathbb{E}[p_\tau] - \mathbb{E}[ z_\tau(S_{\tau-1}) ] \right) \right). $$ 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 treasury in each period. More generally, if the DAO were to commit to a spending path over time, the same result can be obtained despite not spending everything at once. Specifically, suppose that the DAO commits to a declining spending path that takes the form of an exponential decay, or $$ z_t(S_{t-1}) = k S_{t-1} e^{-\lambda(t-1)}, $$ where $0 < k < 1$ is a constant determining the fraction of the treasury spent in each period and $\lambda > 0$ is the decay rate. Then we can state the following result. __Proposition 3__ (Treasury Spending)__:__ _For any given value of $S_0$, there exist a set of critical parameters $\hat{k}$, $\hat{\lambda}$, such that no Type I or Type II equilibrium can arise._ __Proof:__ Consider the forking condition $$ \sum_{\tau = t}^{t^\ast} \left[F\left(b_1^\ast \cdot \frac{1}{\delta^{\tau-t}}\right)\right]^n \geq \kappa (N + t^\ast) - A_t, $$ optimal bid $$ b_t^\ast(h_t) = V^e_t(h_t) + \frac{1}{n} \frac{ F(b_t^\ast(h_t))}{f(b_t^\ast(h_t))} \frac{\partial V^e_t(h_t)}{\partial b_t^\ast(h_t)}, $$ and redemption value under the commited spending path $$ V^e_t(h_t) = \delta^{t^\ast-t} \alpha_{t^\ast} \left( S_{t-1} + \sum_{\tau = t}^{t^\ast} \left( \mathbb{E}[p_\tau] - k S_{\tau-1} e^{-\lambda(\tau-1)} \right) \right). $$ Observe that $$\frac{\partial V^e_t(h_t)}{\partial z_t(S_{t-1})} < 0, \qquad \frac{\partial b^\ast_t(h_t)}{\partial V^e_t(h_t)} > 0, \qquad \frac{\partial \sum_{\tau = t}^{t^\ast} \left[F\left(b_1^\ast \cdot \frac{1}{\delta^{\tau-t}}\right)\right]^n}{\partial b_1^\ast} > 0, $$ which immediately delivers the result. $$\tag*{$\blacksquare$}$$ Proposition 3 shows that by choosing an appropriate spending path, incentives for arbers can be sufficiently reduced to prevent forking. Intuitively, by spending sufficiently quickly, arbers' payoffs by the time they have accumulated enough wins to force a fork are reduced far enough so that their optimal bids become low enough to slow down their speed at which they accumulate 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, they can never get there so that it no longer pays to participate in the auction and only Type III equilibria remain. A similar set of parameters could also be determined at which only Type I equilibria are excluded. In the simulation, for simplicity we tested a linear spending path, i.e. $z_t(S_{t-1}) = q \times \bar{v}$ with $q$ being a fixed parameter set at $q=0$ and q=0.5. <figure style="text-align: center; display: flex; justify-content: space-around;"> <div style="text-align: center;"> <img src="https://hackmd.io/_uploads/r1zUxjzDC.png" alt="Table 13" style="width:95%"/> <figcaption style="font-size: small;"> 13: Mechanism 1 without spending </figcaption> </div> <div style="text-align: center;"> <img src="https://hackmd.io/_uploads/BkRHsJmPR.png" alt="Table 14" style="width:95%"/> <figcaption style="font-size: small;"> 14: Mechanism 1 with 0.5 spending per period </figcaption> </div> </figure> <figcaption style="text-align: center;"><b>Tables 13-14:</b> Equilibrium types for different values of the initial treasury (vertical) and forking threshold (horizontal) under Mechanism 1 and different spending amounts per period.</figcaption> <figure style="text-align: center; display: flex; justify-content: space-around;"> <div style="text-align: center;"> <img src="https://hackmd.io/_uploads/BkWlkTfDR.png" alt="Table 15" style="width:95%"/> <figcaption style="font-size: small;"> 15: Mechanism 3 with P=0 and without spending </figcaption> </div> <div style="text-align: center;"> <img src="https://hackmd.io/_uploads/rk4KyxmDR.png" alt="Table 16" style="width:95%"/> <figcaption style="font-size: small;"> 16: Mechanism 3 with P=0 and with 0.5 spending per period</figcaption> </div> </figure> <figcaption style="text-align: center;"><b>Tables 15-16:</b> Equilibrium types for different values of the initial treasury (vertical) and forking threshold (horizontal) under Mechanism 3 and different spending amounts per period.</figcaption> Under Mechanism 1, the spending path directly influences the share of the treasury that participants receive. Since the allocation is based on contributions, any adjustments in the spending path significantly affect the incentives for arbers. This means that changes in spending can lead to pronounced shifts in behavior and different equilibrium outcomes. Under Mechanism 3, the impact of changes in the spending path is less direct and hence less pronounced than under Mechanism 1. As a result, equilibrium types under Mechanism 3 are less sensitive to these changes.