Mom and Dad are asked to approve of Son’s latest bright idea, whose wisdom is uncertain to both. Son will go ahead only if both parents approve. If the idea turns out well, each parent gains an amount , while if it is bad, each loses an equal amount. If they do not approve, no one gains or loses.
The parents are fully rational, and each has an independent binary (good or bad) signal of the idea’s wisdom, whose value is also binary, and called the state of the world (good = , bad = ). The quality
of each signal is, for each parent, the probability that the signal matches the state of the world. It ranges from, when the signal is uninformative about , to , when the signal deterministically reveals it. Each parent’s quality is independently drawn from a quality distribution whose support is in , and is privately known by the respective parent111The problem is unchanged if the privately-known qualities are merely beliefs. This is because the players are rational and do not entertain conscious biases..
Suppose Dad is asked to make a decision first. His decision is observed by Mom, who decides last. What is the parents’ equilibrium strategy, given a common prior for the state of the world?
This question appears like an elementary exercise in social choice, or in information cascades, or in crowdfunding, but it hides complexity and elements that are novel and absent from the aforementioned fields. Chief among them is social insurance, wherein an early agent (Dad, in this case) relaxes his criteria for participation relying on future agents’ (Mom, in this case) strategies to protect him against loss. Thus, in the classical information cascades model of [Banerjee1992] and of [bikhchandani1992theory]
, Dad would approve strictly based on having a private posterior probability for a good state of the world of at least, as Mom’s future decision is irrelevant for his. In our problem, Dad would participate even if his posterior is below . This is because even if, prima facie, he expects loss, he expects Mom’s future decision to weed out more scenarios where he loses than where he gains. Therefore, his total expectation from participation is positive.
Another element hidden in our small problem we call delegation. This occurs when a player, usually the first (Dad), assumes so much social insurance that his optimal strategy is to participate for any signal or quality, making his decision uninformative. This is not the same as the up-cascade feature of information cascades, where the prior is so favorable that all players participate regardless of type (although up- and down-cascades are also features of our problem). Rather, the prior is not favorable enough for both parents to approve automatically, but is suitable for Dad to delegate the decision to Mom, who, in this case, has to make a solo decision, as Dad’s action carries no information.
We adopt the usual terminology of information cascades, and Bayesian games: A player’s type encompasses all her private information. The likelihood of the state of the world is the ratio . Herding occurs when a player’s optimal action does not depend on her type. A cascade occurs when the same herding behavior persists for all future players. An up-cascade is when the herding is to the action, while a down-cascade is when it is to the action. The type model has the monotone likelihood ratio property (MLRP) (see, e.g., [Herrera2013], [Smith2012]), when inferences from a player’s type on the state of the world are monotonically increasing with the type.
As we will show, we can roll together, for each player, the private quality of a signal in and the private binary signal of , into a new type in that obeys MLRP. We will also demonstrate that the players’ optimal behavior is crucially affected by the range of the quality distribution. We mark by the maximal value in the quality distribution support, and by the minimal value. when there is non-zero probability for a type whose signal deterministically reveals the state of the world. This translates to an MLRP model with unbounded support, which, as usual in unbounded models (see [Smith2012]), precludes herding (because a sufficiently-informed player’s signal can overcome any prior, no matter how good or bad).
We assume , which leads to an MLRP model with bounded support, making herding and cascades possible. This is because the assumption that there exist players who have perfect knowledge of the state of the world appears unrealistic to us. Most questions, and especially predictions (of, e.g., the weather, sporting events, or the success of ventures), have too many unknowns to be answered exactly. This is borne out by empirical studies (e.g., [goel2010prediction], [ban2018all]) that indicate that predictions are limited in their accuracy, even if formed by aggregation of several opinions.
One result emerging from our analysis is that the minimal is also of significance. attaches non-zero probability to a type who is perfectly ignorant of the state of the world. While not impossible, it is often implausible that a player with skin in the game is perfectly ignorant. Setting
sets bounds on that ignorance. This bound also means that every player can trust future players to have a minimally-informative signal. With uniformly-distributed qualities, we show that players delegate only whenexceeds a bound, and never delegate for , i.e., when future players may be perfectly ignorant. If the lower bound and the higher bound are not too close, only early players delegate.
Our Mom and Dad problem is the simplest case of a more general sequential decision problem, with players, the approval (now also called investment) of at least of them needed for a project to take place, with those who approved subject to gain or loss, depending on the project’s outcome. Each player gets an independent good/bad signal on the project’s prospects, with varying quality, i.i.d. drawn from a known distribution, and privately known (or believed) by the player. Players act sequentially, with their actions, but not their signal or quality, publicly observed.
Our Mom and Dad problem represents , while the classical model of [Banerjee1992] and [bikhchandani1992theory] has . The case is a unanimity game, where a project takes place only if all players approve. The general case, , captures sequential fundraising, where potential contributors are sequentially approached for a contribution, and the project takes place if at least of them contribute.
Except for unanimity games, the model differs from voting or social choice games, because, in such, the social choice affects the utility of all players, whatever they voted, while we consider games where only assenters gain or lose. This distinction disappears in unanimity games since there are either no dissenters or no utility.
In this paper we develop a model for the general problem, and formulate equations that are satisfied in equilibrium, then demonstrate interesting economic and game-theoretical aspects of the solution. We seek a Markov Perfect Equilibrium (MPE) of our problem, as defined by [maskin2001markov], where strategies are memoryless and depend only on a payoff-relevant state. We furthermore adopt a refinement that makes the equilibrium unique, except in isolated cases, and show that, in the rest, the optimal strategy is pure, unique, and a threshold one, in which a player invests only if her type is at least equal to a threshold, with indifference to action possible only at the threshold. We demonstrate the cascade limits: Like in the information-cascades setting, an up-cascade occurs at public likelihoods above . Unlike the information-cascades setting, a down-cascade occurs at public likelihoods below Therefore the game is not in a cascade for arbitrarily unfavorable likelihoods, whenever is large enough. The up-cascade limit also places a limit on information aggregation in every sequential fundraising, which we show is public likelihood .
In the general case, , except at cascades, solving for equilibrium analytically is mostly beyond our reach. However, unanimity games , which include the Mom and Dad () problem, are generally solvable. Interesting aspects of the equilibrium include
Depending on the public likelihood, players often, but not always, play the same threshold strategy. In a sense, they divide the “burden” of vindicating the project equally among themselves. We show how this is related to a simultaneous version of the game, and show that players in the sequential game can play unequal thresholds only when the simultaneous game has more than one equilibrium. We provide a necessary condition for two players to play different equilibrium strategies.
For some quality distributions, players never delegate. We give a criterion for this.
For some quality distributions where players may delegate, only the earliest players delegate. In these, a unanimity game starts by zero or more early players delegating by investing regardless of type, in a “reverse” cascade. Unlike in real cascades, this delegation ends before the last player.
In addition, we show that, in unanimity games, players always assume social insurance, i.e., shade their threshold lower, relying on the strategies of future players. We conjecture that this is true for non-unanimity games () too.
1.1 Related Work
Our model is rooted within two lines of literature: (1) the literature of information cascades, and (2) the theoretical study of sequential fundraising.
The information cascades literature originated from the canonical work of [Banerjee1992, bikhchandani1992theory] who studied the question of information aggregation in settings where partly informed agents act sequentially. Agents observe the information revealed by actions taken by previous agents. Their conclusion was that, eventually, the optimal action an agent will take will be identical to the previous action, even when this action contradicts her own private information. This phenomenon is called herding. Additionally, assuming that agents are equally informed, it persists with future agents in a cascade. When herding occurs, the agent’s action is no longer informative, thus future agents can no longer learn. A second question is therefore, what is the probability that the agents herd on the correct alternative? [Banerjee1992, bikhchandani1992theory] studied a model with a binary signal and showed that (1) Herding always occurs in finite time; (2) There is always a positive probability that the agents herd on the wrong alternative. In a follow-up work, [Smith2012] studied a similar model, only with a general signal structure and showed that learning can occur if and only if signals are unbounded in their quality. Subsequent work studied the robustness of these results on various observational structures [Acemoglu2011, Herrera2013], various incentive schemes [Mueller-frank2012, Arieli2018, Moscarini2001a], and even cases where agents are affected by congestion [Eyster2013]. One common feature of these, and the vast majority of existing work, is that the utility of an agent depends solely on actions taken in the past and the information available in the present. We extend this to where agents’ utility is affected by the action of future arrivals. Two rare examples of the introduction of forward-looking agents into an information cascade framework can be seen in [Arieli2018a, Smith2017exp]. In [Arieli2018a], the agents choose between strategically priced alternatives. However, even in [Arieli2018a], the farsighted agents are the firms, while the consumers who potentially herd are myopic. In [Smith2017exp], a social planner assigns actions to agents in order to maximize the discounted sum of agent utilities. To the best of our knowledge, we are first to successfully characterize an equilibrium in the setting of information cascades in which the agents themselves are forward-looking.
The theory of incentives in sequential fundraising processes has been gaining traction with the rise of crowdfunding platforms and venture capital funds. The literature’s lion share focuses on the interaction with the entrepreneur, studying questions of moral hazard, number of investments, and the pace at which investments occur. In [Hellmann2015], the authors study the interplay between early contributors (angels) and later ones (venture capital markets). The authors present a model in which the fundraising process comprises two stages. The initial funding is raised by angels. Following the results from this stage, additional funds are raised from venture capital. The authors use a “costly search model” to find an equilibrium in both markets. In their model, VC investment occurs only if the amount raised from angels surpasses a certain threshold, thus signalling a high-value investment. Angels, who know that the process succeeds only if favorable information is uncovered, have incentives to invest in a priori riskier investments. In equilibrium, both forces are balanced and result in sub-optimal entrepreneur market size. In [Bergemann2004] the authors study the relationship between an investor and an entrepreneur. They present a repeated moral hazard game in which an entrepreneur raises funds from an investor. After raising an investment, the entrepreneur decides which part of it is allocated to the company and which is diverted to her own goals. The authors compare between two scenarios, one where investors can monitor the entrepreneur decision and one where they cannot. The lack of information creates a scenario in which the investor is implicitly committed to a stopping time, which is invariant to the entrepreneur interim behavior. In [Bergemann2008], investors decide sequentially over the pace at which a venture is funded. They find that in this scenario, investment pace starts slowly and accelerates with favorable information. The work closest to ours is [Halac2018]. The model presented in [Halac2018] comprises agents who have heterogeneous investment size, and act sequentially and decide whether to invest in a firm. The firm can alter its revenue stream, which determines the investment attractability. They show that the optimal firm strategy increases the inequality between the returns of large investors and smaller ones. We supplement [Bergemann2008, Bergemann2004] by requiring investments to initiate the process, thus investors cannot alter their decision based on interim output. We extend [Hellmann2015] to a case of more than two rounds of investment. We supplement [Halac2018] as in our model, agents are aware of the exact amount of participation required, and thus can make inferences from the decision of the pivotal investor, who, in turn, knows she is pivotal. We show that the pivotal nature of the latter plays a major role in her decision and on the efficiency in which information is aggregated, and allow for both social insurance and delegation which were unobserved in previous work.
Another related line of work is the literature that studies crowdfunding [Alaei2016, Strausz2017, arieli2018one]. In [Strausz2017], the author studied the robustness of crowdfunding campaigns to moral hazard and charaterizes the conditions under which crowdfunding campaigns are resilient to entrepreneurial abuse. Alaei et. al. [Alaei2016]
presented a model where agents’ actions depend on past actions and also on their estimation of what future agents will do. Their setting, motivated by crowdfunding campaigns, has agents with private valuations, and the public information is solely the number of adoptions. We present a model where agents have a noisy signal on quality. In[Alaei2016]’s model, the risk facing consumers is to invest in a failed campaign due to contribution costs. In ours the risk is due to a poor product. Due to this, our model can be applied to other cases, such as sequential voting and group buy. [arieli2018one] presented a static model of crowdfunding, in which agent signals are binary, and show that agent participation is affected by the decision of their colleagues, when the population size is sufficiently large. They use the term “social insurance”, roughly in the meaning used in this paper. To the best of our knowledge, this is the first dynamic model that studies this aspect, and gives results on equilibrium strategies, efficiency, and even yields counterintuitive results. This is an addition to previous literature and is empirically supported in crowdfunding [Mollick2014].
The paper is organized as follows. Our model is presented in Section 2. Section 3 analyzes and solves the problem up to an equilibrium characterization. In Section 4 we demonstrate properties of the solution in cases of interest and in the general case. We offer conclusions and general remarks in Section 5.
players, labeled , need to take a joint decision on a project. Each has two possible actions, “invest” and “decline”. Player makes a public decision first, followed by player . Player ’s action is marked . The project takes place only if at least players invest, i.e., if .
The state of the world, , is either (project is “good”), or (project is“bad”).
If the project does not take place, all player utilities are zero. If the project does take place, each player who invested gets utility , while players who declined have zero utility.222A rescaling of likelihoods will handle the case where upside and downside are not equal.
Each player receives a private, noisy signal of . Signals are mutually independent, conditionally on . The probability that the signal is correct () is individual to the player, called the signal quality and marked . With probability , the signal is wrong (). Qualities are i.i.d. samples from a distribution where . The distribution has piecewise-continuous density with no atoms, and , . Each player privately knows her quality.
The prior likelihood of the state of the world, marked , is publicly known. It is defined
A Markov perfect equilibrium (MPE) of the game is sought. This means a perfect Bayesian equilibrium (PBE) in Markovian strategies, as defined by [maskin2001markov]. The strategy of each player
is accordingly a probability distribution over the action space, which is a function of the payoff-relevant state , and of the player’s privately-known type , defined below, the state comprising
: The current public likelihood of .
: The number of investments still needed for completion.
: The number of players still to make a decision. Note that this encodes the player’s identity .
In accordance with the Markov property, each state may be considered the starting point of the fundraising, in which the initial state is irrelevant.
A player’s type consists of her private information, i.e., the pair . For each player , we roll this pair into a single real number , defined
so that . Note that ’s support is .
In addition, we make the following tie-breaking refinement:333Despite some similarities, our refinement is not Selten’s Trembling-Hand Perfection. Similar tie-breaking mechanisms have been used in the context of dynamic analysis. See, for example, the Obfuscation Principle presented in [Ely2017]. There, a social planner declares agent strategies before the game commences. When a player has several strategies that are tied for having optimal utility expectation, the player must play a mixed strategy in which at least two of the tied strategies are played with a probability not less than , a fixed parameter, that is positive but arbitrarily small.
The history of player , is the list of actions of previous players ( is an empty list).
Mark by the public likelihood of the state of the world after player ’s action, inferred from and , i.e.
The private likelihood of a player, given her privately-known type (as the type and the history are mutually independent), is
We will show (Corollary 3) that, except for irregular states, defined below, there is a unique MPE, and equilibrium strategies are pure threshold strategies, i.e., players invest if their type is larger than the threshold, decline if below it, and are possibly indifferent if their type is equal to the threshold. The strategy is therefore represented by a function444As no two players can be in the same state, we use a common strategy function .
where if and if .
3.2 The Case : Information Cascades
When we are in a situation similar to that described in the classical model of [Banerjee1992, bikhchandani1992theory]: The project has already been decided on, or will be if the current player invests. Therefore the behavior of future players has no bearing on the current player’s strategy: There is no social insurance, and the player decides strictly based on her private likelihood (3), investing when it is and declining when it is .
Proposition 1 ().
If the number of investments needed is reached or exceeded, the probability for completion is . A player with type has expectation for investing
which is non-negative iff , i.e., when . ∎
3.3 Probability of Completion
We will show that a player’s optimal strategy critically depends on the probability of completion, which we now define.
In an -player game, let be a profile of strategies in MPE for each player.
Define the probability of completion, as the probability that the project takes place (i.e., the fundraising completes) when the state of the world is and the players play their strategies in , given public likelihood , outstanding investments and remaining players. Clearly, when and when , for any and .
These are examples of states where the probability of completion does not depend on the strategy profile. In general, if there is a unique MPE, there is a unique probability of completion, which we mark simply . We shall later derive a recurrence equation for it in Corollary 2.
States where there are multiple MPE’s are called irregular, and are excluded from our discussion. We later (Section 3.8) characterize when a state is irregular, and show why it is reasonable to not consider them.
Let be some Markovian strategy that the first player plays in state . It depends on the player’s private type. Other players, not knowing the first player’s type, make inferences based on their knowledge of . Let the type-independent probability that the player declines under and state of the world be marked :555In a slight abuse of notation, we shall later (Section 3.5) intentionally reuse as the c.d.f. of the type distribution, once we have proven that the only strategies that need be considered are threshold strategies. By Bayes’ rule, if the player invests, the posterior likelihood changes to
and the state changes to , while if the player declines, the posterior likelihood changes, by Bayes’ rule, to
and the state changes to .
If neither of these two states are irregular, there is a well-defined probability of completion for every possible action of the player. Thus, the probability of completion under , which we mark , is well-defined, equalling
3.4 Equilibrium Strategies are Threshold Strategies
We now prove that the only strategies that are played in MPE are threshold strategies.
For , we define as a strategy with threshold , investing for every type , declining for every type , and possibly indifferent for type .
Theorem 1 (Threshold Strategy).
Excluding irregular states , let be an MPE with public likelihood , outstanding investments, and remaining players. Then is a threshold strategy , where is not necessarily a type in .
Let a player have a type with public likelihood , outstanding investments and remaining players. Her private posterior likelihood is, by (3), , and her probability for is .
Let the type-independent probability for declining under be . The posterior likelihood from investing, , is given in (5). is independent of , since the type is private and unobserved. So, If the player invests, her expectation, marked , is
is non-negative iff
The right-hand side of (9) does not depend on type , while the left-hand side increases with increasing , and varies continuously from to when varies from to . It follows
By the mean-value theorem, there exists a where (9) holds with equality.
If calls for declining at any , the player will deviate, as , and so the strategy is not playable in equilibrium.
If calls for investing at any , the player will deviate, as , and so the strategy is not playable in equilibrium.
Therefore the only strategies playable in equilibrium are threshold strategies where for (9) holds with equality. ∎
3.5 Inferences from Actions in Threshold Strategies
At equilibrium, by Theorem 1, all players play threshold strategies, which are commonly known. A player’s type is not observable, but the fact that she invested proves that her type is greater than her threshold, while if she declined, the inference is that her type is below the threshold. Since these events have, a priori, different conditional probabilities for each state of the world , observing the action leads, by Bayes’ rule, to an updated public likelihood of .
We derive from , the quality density function, the distribution of type
, a random variable with support in, conditional on , with density and c.d.f. . By (1) and the definition of the quality
regardless of , and is monotonically increasing in , so type distributions always have the MLRP property.
For the undefined is taken to be the limit by L’Hôpital’s rule
For , the undefined is taken to be the limit by L’Hôpital’s rule
The following lemma will be useful.
and are monotonically increasing in .
See Appendix. ∎
Since we deduce from Lemma 1 first-order stochastic dominance
with equality only when .
Thus, observing an investment () by player , whose threshold strategy is , when the prior public likelihood is
, and the prior probability foris , we derive by Bayes’ rule
From which we conclude that the posterior public likelihood inferred from investment is
Similarly, the posterior public likelihood inferred from a decline is
Due to stochastic dominance (15), we have, for every
Corollary 1 (Threshold Indifference Condition).
In equilibrium, a threshold strategy satisfies
We now derive a recurrence equation for the probability of completion .
Corollary 2 (Probability of Completion).
Except at irregular states , the probability of completion is continuous in , equalling
3.6 The Equilibrium Strategy
In previous sections we show that, in MPE, a player plays a threshold strategy where the threshold satisfies (19). If (19) has a single solution , this single solution is the player’s strategy in MPE. But, If there are multiple ’s satisfying (19), she must choose.
Proposition 2 (Optimal Strategy).
In MPE, a player’s strategy is a Markovian strategy whose type-independent utility expectation is not exceeded by any other Markovian strategy.
In a general game, in MPE a player plays a strategy that is a function of the payoff-relevant state, and is a best response to the strategies of all concurrent and future players. The strategy does not depend on the strategies of past players, as their strategies, to the extent that they are payoff-relevant, are already encompassed in the state.
In the sequential fundraising a player has no concurrent players. A playable strategy is a strategy from which the player will not deviate for any type. The player’s optimal choice is the playable strategy that has maximal utility expectation before she knows her type, given the state and the strategies of future players. No simultaneous change of strategy by future players is implied by this strategy choice, since, for future players the current player is a past player whose strategy does not affect theirs.666This argument would not work in an SPE/PBE solution concept, where a player’s strategy is a best response to all other strategies, past or present. ∎
The type-independent utility expectation is given by the following proposition.
Proposition 3 (Utility Expectation).
Excluding irregular states , the type-independent utility expectation for a strategy is
The expectation for declining is , so is the expectation for the remaining action, investing. The probability that the player invests is , and so, when , which occurs with probability , the player gains with probability
Similarly, when , which occurs with probability , the player loses with probability
3.7 Equally-Optimal Strategies
When a player has several strategies with maximal type-independent utility expectation, we show that the player nevertheless prefers one of these strategies, when we assume, according to our model’s refinement, that the player always mixes between these strategies.
Proposition 4 (Tie-Breaking Equally-Optimal Strategies).
Excluding irregular states , given strategies with equal type-independent utility expectation, a player who mixes between the strategies, assigning probability at least to at least two of them, maximizes her utility expectation by assigning maximal probability to some that maximizes the discriminator
See Appendix. ∎
3.8 Equilibrium Characterization and Irregular States
We summarize the above sections with a characterization of the MPE of a sequential fundraising.
Corollary 3 (Equilibrium Characterization).
Except at irregular states , at public likelihood , investments outstanding and players remaining, there is a unique pure MPE in which, in every subgame, all players play threshold strategies. The playable threshold strategies are those that satisfy (19), and the MPE strategy is the one with a highest type-independent utility expectation, as given in (3), where, in case of equality in utility expectations, the strategy with highest value of (22) is played in MPE. The state is irregular if (22) fails to discriminate between two or more equally-optimal strategies in any subgame.
As the corollary states, an irregular state is one where any subgame has multiple MPE’s. Consequently, the following is a recursive characterization of these states.
Corollary 4 (Irregular States).
The state is irregular
Never, for .
If it has multiple MPE’s, with several equally-optimal strategies whose discriminators (22) are equal.
If it has an MPE where the first player plays , and any of the states or is irregular.
The irregular states ultimately depend on the quality distribution , but this characterization shows why they are most probably rare, finite in number for any bounded , and often do not occur at all, though this may be hard to prove.
4 Solution Properties
4.1 Herding and Cascades
Herding describes a situation where all types have the same optimal action. A cascade occurs when herding persists among players, so that when a cascade starts, all future players decline (down-cascade) or all invest (up-cascade).
An up-cascade occurs wherever the public likelihood is at least .
Theorem 2 (Up-Cascade).
For every and , and an up-cascade is in progress iff .
An up-cascade occurs iff the probability of completion is . We prove that when by induction on . For this is true for any . Assume the theorem true up to . Then, a threshold strategy must satisfy, by (19) and the induction hypothesis . So
Thus . ∎
At the opposite end, for a herd of decliners. Since the expectation of such a strategy is , it can be optimal only if it is unique. Thus from (19)
Corollary 5 (Herd Decline).
, i.e., declining is optimal for all types iff
In a down-cascade, all remaining players decline. Using Corollary 5 repeatedly we get
Corollary 6 (Down-Cascade).
For every and , and a down-cascade is in progress iff .
For herding on investments, a necessary condition is (19) with . It is not sufficient, as it may not have maximal utility expectation.
Corollary 7 (Herd Invest).
4.2 Information Aggregation
Information aggregation, or learning, occurs when the state of the world becomes known with higher probability, or likelihood. So we ask, what is the public likelihood , at completion of the fundraising?
To bound , we observe that information aggregation stops in an up-cascade. Therefore, the public likelihood before the last player’s action is bounded above by , the up-cascade limit by Theorem 2. The inference from the last player’s investment is bounded by , by (13) and Lemma 1. It follows
Corollary 8 (Learning Bound).
Let . In equilibrium, .
4.3 Example: Uniformly-Dense Qualities
The simplest assumption we can make about the distribution of qualities is that they are uniformly distributed between their lower bound and upper bound . Here, we work out the resulting distributions and player behavior in some sequential fundraising games.
Let qualities be uniformly dense, i.e., for
We use this example distribution to demonstrate several facts:
In the information cascades setting (), some quality distributions allow cascades to start, while in others, if the prior is not a priori a cascade (i.e. if ), a cascade never starts. We shall later (Section 4.4.3) show that this property is intimately tied to delegation in sequential fundraising.
Proposition 5 (Minimum for Cascades).
When qualities are uniformly distributed , cascades cannot start in the information cascades setting () when . For each there is a minimal , where cascades are possible iff .
See Appendix. ∎
We plot solutions of the Mom-and-Dad problem () for this uniformly-distributed qualities with (Figure 4), (Figure 4) and (Figure 4). We also plot the solution for players () with (Figure 4). Since in most of the range, the players play equal thresholds, represented as colored bars, the plots appear to have a color mixture effect.
All thresholds below (herd invest, or delegate) are equivalent, as are all thresholds in the range (invest on ), since they fall outside the type support . Their different values reflect a different point of indifference in (19).
In the first plot, , the two parents always play the same threshold strategies, and no delegation ever takes place. In the 2nd and 3rd, and , the parents play different thresholds in part of the public likelihood range. For , Dad always plays a threshold equal or lower to Mom’s (and so is the only one to delegate), while for , both parents may play the lower threshold, and both may delegate.
The threshold strategy can be non-monotonic. This is demonstrated by (Figure 4), where, as the plot shows, the first player (Dad) has a non-monotonicity in his threshold at .
4.4 Unanimity Decision
In this section we consider equilibrium behavior in unanimity games.
4.4.1 Equilibrium Characterization
The equilibrium characterization given in Corollary 3 takes a special form in unanimity games.
Proposition 6 (Equilibrium Characterization).
Let . Except at irregular states , let be the thresholds of the MPE strategies for each player. Then,
The probability of completion is
and for every
The strategies have equally-maximal utility expectation
with having the largest discriminator, which, for is
(26) follows from the fact that the fundraising completes iff all players invest.
For each , , , is playable by the first player, because the threshold indifference condition (27) holds under every permutation of the thresholds. Furthermore, expectation (3) for the unanimity game is (28), due to (26), and is symmetric under permutation of the thresholds, so every has the same utility expectation. This symmetry is broken by the discriminator (22), which, in the unanimity game takes the form (29), due to (26). ∎
4.4.2 Social Insurance
We show that players in unanimity games always assume social insurance, meaning that, given the same public likelihood, a player shades her threshold lower than would be played in a single-player game (the information-cascades setting with ), i.e., that for every . Indeed, we get the following, stronger result,
Theorem 3 (Social Insurance).
Let , and let be the strategy in MPE of player with prior likelihood . In equilibrium with equality only for the last player or in a cascade.
An extreme form of social insurance is delegation, defined as herding on investment when not in a cascade. The following theorem summarizes properties of delegation.
Theorem 4 (Delegation).
Let . Except at irregular states , let be the thresholds of the MPE strategies for each player. In equilibrium,
In a quality distribution , delegation takes place, for some , iff there exists for which
which is also the necessary and sufficient condition that, under , up-cascades can start in the information-cascades setting.
If any player delegates (has threshold when ), then for all players , .
In a quality distribution , only the earliest players delegate iff for every satisfying (31)
In a quality distribution , only the latest players delegate iff for every satisfying (31)
See Appendix. ∎
For uniformly-distributed qualities (see Section 4.3), Theorem 4, in conjunction with Proposition 5, shows that players never delegate for (exemplified in Figure 4), but may delegate starting at some minimum (as shown in Figures 4-4). This means that, at least for uniform distributions, players delegate only when they can expect future players to have sufficiently accurate signals.
Quality distributions where delegation is by early players only (following Theorem 4(3)) are common (shown in 3 out 4 of Figures 4-4), and exhibit a time reversal of sorts: While in the information cascades setting an up-cascade might start after several players invested, in the sequential fundraising game players pre-empt and avert this possibility by delegating. Whenever a player’s approval with given threshold would trigger an up-cascade in the information cascades setting, in our fundraising setting all previous players would delegate before such a strategy player. Furthermore, they have “reverse” cascades, wherein herding takes place from some player backwards.
Another consequence is that up-cascades cannot start before the fundraising completes.
4.4.4 When are Thresholds Equal?
Consider the unanimity game played simultaneously rather than sequentially. All players are in a perfectly-symmetrical situation: They need to take a decision, based solely on the prior public likelihood . The project will take place iff all players simultaneously approve.
Therefore there exists a symmetric MPE in the simultaneous game, in which all play strategies with the same threshold . This equilibrium may not be unique.
The sequential game is apparently different, in that players observe the decisions of others and adjust their action accordingly. In fact, in a unanimity game, this difference is an illusion: Every player may assume that others invested, since, if they did not, their own decision does not matter. Observing that others, in fact, invested, changes nothing.
This shows why every MPE of the sequential game is an MPE of the simultaneous game (but not vice versa, as Proposition 2 is valid only for the sequential game).
This explains why players may play different thresholds, but it also explains why often all or some play the same strategy: When the symmetric equilibrium is the only MPE of the simultaneous game, it must also be the only MPE of the sequential game.
In fact, we can formulate a necessary condition for players to play different strategies.
Proposition 7 (Equal Thresholds).
In MPE, two players play unequal strategies only if is not strictly monotonic between and . Equivalently, this can happen only when
changes sign between and .
See Appendix. ∎
4.5 The General Case: Non-Unanimous Decisions
In the general case, , the fundraising will succeed if at most contributors decline to invest. There are therefore subgames with a positive probability for completion.
Based on the equilibrium characterization (Corollary 3), a solution for equilibrium requires the solution of simultaneous equations, one for each subgame, each formed as (19). In Section 1 we provided a complete solution of the case , and in Section 4.4 we provided a thorough analysis of the case . A similar treatment of is at present beyond our reach, though we have provided a characterization of its cascades (Section 4.1), and a recurrence relation for the probability of completion (Section 3.3).
A numerical solution for equilibrium strategies must therefore rely on dynamic programming.
A stumbling block in the analysis is our inability to prove the following, which we conjecture.
Conjecture 5 ().
Except at irregular states , for every public likelihood , investments outstanding and players remaining
with equality only in a cascade or for .
Conjecture 5 has an immediate consequence on social insurance, namely that in every state in a sequential fundraising players assume social insurance. Indeed, wherever the conjecture is true, (19) translates into , with equality only for or in a cascade. On the other hand, with the same public likelihood, the last player’s equilibrium strategy satisifies . Therefore .
Corollary 11 (Social Insurance).
Let be the strategy of player in MPE with prior likelihood , investments outstanding and players remaining. If Conjecture 5 holds, i.e., if