Symmetric Bayesian Nash Equilibrium Research Articles

Social Sourcing: Incorporating Social Networks Into Crowdsourcing Contest Design

In a crowdsourcing contest, a principal holding a task posts it to a crowd. People in the crowd then compete with each other to win the rewards. Although in real life, a crowd is usually networked and people influence each other via social ties, existing crowdsourcing contest theories do not aim to answer how interpersonal relationships influence people's incentives and behaviors and thereby affect the crowdsourcing performance. In this work, we novelly take people's social ties as a key factor in the modeling and designing of agents' incentives in crowdsourcing contests. We establish two contest mechanisms by which the principal can impel the agents to invite their neighbors to contribute to the task. The first mechanism has a symmetric Bayesian Nash equilibrium, and it is very simple for agents to play and easy for the principal to predict the contest performance. The second mechanism has an asymmetric Bayesian Nash equilibrium, and agents' behaviors in equilibrium show a vast diversity which is strongly related to their social relations. The Bayesian Nash equilibrium analysis of these new mechanisms reveals that, besides agents' intrinsic abilities, the social relations among them also play a central role in decision-making. Moreover, we design an effective algorithm to automatically compute the Bayesian Nash equilibrium of the invitation crowdsourcing contest and further adapt it to a large graph dataset. Both theoretical and empirical results show that the new invitation crowdsourcing contests can substantially enlarge the number of participants, whereby the principal can obtain significantly better solutions without a large advertisement expenditure.

(A)symmetric equilibria and adaptive learning dynamics in small-committee voting

We study equilibrium selection in a common-interest voting model with three alternatives. In the model, symmetric Bayesian Nash Equilibria (BNE) of varying efficiency are known to exist. Employing evolutionary adaptive learning simulations, we find interesting new equilibria. In simulations, we distinguish between individual learning (agents learn from their own experience) and social learning (agents may also imitate each other’s strategies). We also vary whether voters are randomly re-matched. Social learning consistently converges to steady states that match efficiency-maximizing symmetric BNE. Individual learning with fixed matching converges to, on average, more efficient steady states, which we confirm as pure-strategy asymmetric BNE. This class of BNE has received little attention in the literature. We show that these BNE may be more efficient and discoverable through an adaptive learning process.

A Blockchain Enhanced Coexistence of Heterogeneous Networks on Unlicensed Spectrum

Due to the forecasted fast increasing cellular traffic and the already highly congested licensed spectrum, it is critical to exploit and utilize the unlicensed spectrum resources for the fifth-generation (5G) and beyond networks. A challenging problem is the coexistence of 5G and other networks with fair, reliable, and efficient sharing of the unlicensed spectrum. In this paper, we propose a blockchain-enhanced distributed spectrum sharing scheme for coexisting multiple operators and multiple WiFi APs. We design a novel lightweight and efficient consensus mechanism, named Proof of Strategy (PoG). In this consensus mechanism, the problem of spectrum sharing is used as a consensus puzzle, and the part of the unlicensed spectrum is used as the ‘fee’ of miners. With such a design, the computing overhead of the consensus process is expected to be reduced significantly. We develop a non-cooperative game to analyze the behavior of the miners and obtain a symmetric Bayesian Nash equilibrium under the uniform distribution of mining cost estimation. It can be found mathematically and experimentally that the strategy of the winner tends to maximize the system revenue by sharing the unlicensed spectrum resource. Furthermore, to reduce the impact of heavy interactions on system throughput, the operation of WiFi APs in the proposed scheme can be adaptively switched between ‘contention mode’ and ‘blockchain mode’ according to the network traffic load. The dynamic behavior is constructed as an evolutionary game, and the existence and uniqueness of equilibrium points are proved by theoretical analysis. Simulations demonstrated the fairness and effectiveness of the proposed blockchain-based scheme and the mode switching method for distributed spectrum sharing by heterogeneous wireless networks.

Corporate Bonds: Uniform Auction under Risk and Budget Limits

A perfectly divisible corporate bond is allocated to a set of bidders characterized by limits both to their budget, but most importantly to the risk entailed in their portfolio. Bidders possess symmetric information concerning the secondary market's yield. We choose to use a uniform pricing mechanism contrary to discriminatory as the former generates more revenues and reduces the winner's curse. As a first step, we prove the existence of symmetric Bayesian Nash equilibrium when risk-neutral bidders respond to an exogenous secondary market providing the necessary comparative statics. In the second stage, we confirm the existence of equilibrium when bidders' types are affiliated with the secondary market by a copula.

War of Attrition with Incomplete Information and Fuzzy Players' Types

The result on existence of a pure-strategy symmetric Bayesian Nash equilibrium in the war of attrition is generalized for fuzzy players’ actions and types.

On the consumer problem under an informational externality

We use the Hendricks and Kovenock (RAND J Econ 20(2):164–182, 1989) framework to study the consumer problem under an informational externality. The informational externality arises when each consumer of a social network is endowed with private information regarding the quality of a good. In such situations, the past purchasing decisions of the consumers are informative and, thus, are used as partially revealing signals of private information. Asymmetric information and the observability of actions render the consumer problem dynamic and strategic because the purchasing decision of a consumer affects the other consumers’ future payoffs through the learning process. We show that there exists a unique symmetric Bayesian–Nash equilibrium. The informational externality increases the likelihood for a consumer to refrain from purchasing the good immediately to make a more informed decision in the future.

Equilibrium in a war of attrition with an option to fight decisively

We develop a symmetric incomplete-information continuous-time two-player war-of-attrition game with an option to fight decisively. We show that there exists an essentially unique symmetric Bayesian Nash equilibrium. Under equilibrium, the game does not end immediately, and a costly delay persists even with the availability of the fighting option that ends the game if chosen. In addition, there exists a critical time in which a fight occurs unless a player resigns before that time.

AUCTION–LOTTERY HYBRID MECHANISMS: STRUCTURAL MODEL AND EMPIRICAL ANALYSIS

AbstractThis article studies an auction–lottery hybrid mechanism that is widely adopted in allocating new vehicle licenses in China. We characterize individuals' entry and bidding strategies in a symmetric Bayesian Nash equilibrium, structurally estimate individuals' value distribution from a data set of a Guangzhou program, and evaluate the performance of the mechanism. Based on the estimated distribution and counterfactual analysis, our study suggests that a hybrid mechanism preserves 83% efficiency and 52% of revenue, while it improves equity by 25 times compared to a pure auction. We show that allowing auction losers to participate in the lottery can further enhance the performance.

Auction-Lottery Hybrid Mechanisms: Structural Model and Empirical Analysis

This article studies an auction–lottery hybrid mechanism that is widely adopted in allocating new vehicle licenses in China. We characterize individuals' entry and bidding strategies in a symmetric Bayesian Nash equilibrium, structurally estimate individuals' value distribution from a data set of a Guangzhou program, and evaluate the performance of the mechanism. Based on the estimated distribution and counterfactual analysis, our study suggests that a hybrid mechanism preserves 83% efficiency and 52% of revenue, while it improves equity by 25 times compared to a pure auction. We show that allowing auction losers to participate in the lottery can further enhance the performance.

Strategies for Making Best Offers on eBay

We model for “Buy-It-Now or Best Offer” auctions on eBay using two different models. In the first model, risk-neutral bidders submit bids in serial and try to surpass a stochastic seller threshold while taking into account how many previous failed bids were made by other bidders. We compute optimal strategies for this model and show that bidder expected surplus decreases in the number of previous failed bids. In the second model we assume bidders do not know how many previous failed bids have been made, and instead use a first-price sealed-bid mechanism with a buy-out price where bidders serially submit bids with the knowledge that no previous bidders have used the buy-out price. We derive a unique equilibrium bidding strategy for risk-neutral bidders in this serial model, show that any equilibrium in a similar parallel bidding model is the same as the equilibrium in the serial model, and compute seller revenue. In particular, under certain circumstances, bidders will bid more in this format than they would in a standard first-price sealed-bid auction, but that a seller maximizes expected revenue by setting a buy-out price higher than any bidder is willing to pay thereby making the auction essentially a first-price auction. KEYWORDS: Auction Theory; eBay; Buy-It-Now or Best Offer; Symmetric Bayesian Nash Equilibrium; Buy-Out Price; First-Price Sealed-Bid

Come on in, the Water's Fine! An Experimental Examination of Hybrid IPO Auctions with a Public Pool

We offer experimental and theoretical evidence that the auction method for initial public offerings (IPOs) may be improved through the use of hybrid auctions with separate retail tranches or ‘public pools’. Such hybrids, which combine a price-setting tranche (an auction or book building) with a separate tranche that allows investors to place orders without specifying a price (the public pool), have been used for IPOs around the world. We develop theory, then run laboratory experiments to examine the effects of a public pool on multi-unit uniform price auction IPOs. Our experimental auction design incorporates key features of the IPO process such as endogenous bidder entry, costly information acquisition, differing capacity constraints and uncertainty with respect to the intrinsic value. Simulations are used to characterize the Symmetric Bayesian Nash Equilibria (SBNE) for both pure and hybrid auctions in a model that is calibrated to key parameters from our experimental data, generating predictions for the remaining variables. As predicted, a public pool tranche improves auction performance by increasing proceeds, lowering price volatility, reducing price error and reducing the incentive for small bidders to free ride by submitting extremely high bids. Underpricing occurs in both treatments but is less severe with the public pool. We also show that in collusive-seeming multi-unit auction equilibria, it may be optimal for informed, rational bidders to place clinching bids strictly above the expected value per unit, leading to very steep demand curves. Overall, our results imply that both IPO auctions and crowdfunding may be improved by restricting retail investors to a separate, non-price-setting tranche.

On the Consumer Problem under an Informational Externality

We explain the Hendricks and Kovenock (1989)'s framework by studying the behavior of two strategic firms under an informational externality. The informational externality arises when each firm of a social network is endowed with private information regarding the profitability of the investment. In such situations, the past decisions of the firms are informative and, thus, are used as partially revealing signals of private information. Asymmetric information and the observability of actions render the firm's problem dynamic and strategic because the investment decision of one firm affects the other firms' future payoffs through the learning process. We describe the model and we show that there exists a unique symmetric Bayesian Nash equilibrium. The informational externality increases the likelihood for a firm to refrain from investing immediately in order to make a more informed decision in the future.

Strategic Interaction in a Stock Trading Chat Room

We consider a model of an internet chat room with free entry but secure identity. Traders exchange messages in real time of both a fundamental and non-fundamental nature. We explore conditions under which traders post truthful information and make trading decisions. We also establish a symmetric Bayesian Nash equilibrium in which momentum traders pro t from their exposure to informed traders in the chat room. The model generates a number of empirical predictions: (1) the non-skillful traders follow the skillful traders; (2) the more skillful traders are more frequently followed by others; (3) the non-skillful traders bene t from following. We test and con rm all three predictions using a data set of chat room logs from the Activetrader Financial Chat Room.

A NOTE ON DISCRETE BID FIRST-PRICE AUCTION WITH GENERAL VALUE DISTRIBUTION

This paper evaluates the discrete bid first-price sealed-bid (FPSB) auction in a model with a general value distribution. We show that a symmetric Bayesian Nash equilibrium exists for the discrete bid FPSB auction. We further prove that the discrete bid FPSB equilibrium conditionally converges to that of a continuous bid FPSB auction.

Preemption Games: Theory and Experiment

Several impatient investors with private costs Ci face an indivisible irreversible investment opportunity whose value V is governed by geometric Brownian motion. The first investor i to seize the opportunity receives the entire payoff, V-Ci. We characterize the symmetric Bayesian Nash equilibrium for this game. A laboratory experiment confirms the model's main qualitative predictions: competition drastically lowers the value at which investment occurs; usually the lowest-cost investor preempts the other investors; observed investment patterns in competition (unlike monopoly) are quite insensitive to changes in the Brownian parameters. Support is more qualified for the prediction that markups decline with cost. (JEL C73, D44, D82, G31)

Informative voting and condorcet jury theorems with a continuum of types

We consider a model of information aggregation in which there are two possible states of the world and agents receive private signals from the set of probability measures over the binary state space – the unit interval. For a reasonably general set of signal densities, a unique symmetric Bayesian Nash equilibrium in responsive strategies exists and voting is informative in this equilibrium. Asymptotic analysis shows that society makes the correct decision almost surely as population size grows. In contrast to findings of Feddersen and Pesendorfer (1998) in the finite signal space case and Duggan and Martinelli (1999) in an alternative model in which the signal space is a continuum, this result holds for unanimity rule. The key to the efficiency of unanimity rule is that there are perfectly informative (or at least nearly perfectly informative) signals. A corollary to the asymptotic efficiency result is that for all rules the collective performs better than a single agent's dictatorship for large but finite populations. This need not be true for arbitrary population sizes.

Public Firms as Regulatory Instruments with Cost Uncertainty

We re-examine the regulatory role of a public firm in an environment of private but correlated information about irndustry costs. We study three regimes of mixed-market interaction involving both public and private firms: a symmetric Bayesian-Nash equilibrium, an asymmetric Bayesian equilibrium in which the public firm is able to commit to production before the private firms, and a mechanism in which the regulator designs an incentive- compatible schedule for the industry. We find that a public firm plays an important strategic informational role which strengthens its role as a disciplinary regulatory instrument. Further,