CC BY
0The Partial Solution of First Price Sealed Bid Auction in the Scope of Mechanism Design Theory and With Uniformly Distributed Types
Petrosyan Rafayel M.
Phd Student at Public Administration Academy of Republic of Armenia,
Faculty of Management (Yerevan, RA) https://orcid.org/0009-00Q2-2218-4536 rafa.tgm@outlook.com
UDC: 339.1; EDN: ZAQZYS; JEL: D44, C57; DOI: 10.58587/18292437-2024.3-170
Keywords: Mechanism design theory, mechanism, auction, first price sealed bid auction, dominant strategy, Bayesian Nash equilibrium state, incentive compatability, social choice function
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Частичное решение закрытого аукциона первой цены в области теории дизайна
механизма и с равно распределенными типами
Петросян Рафаел М.
Аспирантр кафедры управления, Академия государственного управления Республики Армения (Ереван, РА)
Аннотация. В статье анализируется аукцион первый цены в рамках теории конструктирования механизмов и с равномерно распределенными типами. Было найдено частичное решение для аукциона первой цены, состоящий из двух агентов. Оценены стратегии байесовского равновесия Нэша, состояние равновесия и реализуемая функция социального выбора для аукциона. Механизм был разработан для правдивой реализации функции социального выбора для аукциона, состоящего из двух агентов. Были оценены такие переменные, как вероятность победы агента, ожидаемая полезность агента и ожидаемый доход продавца.
Ключевые слова: Теория дизайна механизма, механизм, аукцион, аукцион с закрытыми предложениями по первой цене, доминирующая стратегия, состояние байесовского равновесия по Нэшу, совместимость стимулов, функция социального выбора
Introduction
Mechanism design theory has widespread applications across various fields of economics. The exploration of mechanism design and its market implementations has attracted considerable attention from researchers. The primary aim of mechanism design theory is to develop mechanisms that achieve pre-defined objectives.
Mechanism design operates within the constraints of informational asymmetry and individual rationality. In this context, agents act in their self-interest, striving to maximize their utility. Meanwhile, the mechanism designer aims to achieve desirable social and economic outcomes for all parties, guided by specific criteria of interest [1, pp. 1-19], [2, pp. 1-8].
Particularly notable is the theory's efficiency in auction theory. Mechanism design theory facilitates the analysis and comparison of different auction types, assessing the efficiency of their implementation and application. Leveraging this theory, new auction models have been developed, whose equilibrium states align with the mechanism designer's initial goal function [3, pp. 1-62]:
The goal of this article is to analyse the first price sealed bid auction consisting of two agents within the scope of mechanism design theory.
In order to achive the goal the following objectives will be addressed:
• analyse the first price sealed bid auction consisting of two players using the methodology and tools provided by mechanism design
• identify the Bayesian Nash equilibrium strategies of the agents and the Bayesian Nash equilibrium state of the auction,
• discover the social choice function that is implemented by the mechanism of first price sealed bid auction,
• assess the winning probability of agents implementing Bayesian Nash equilibrium strategies based on types,
• calculate expected utilities of agents and the expected revenue of the seller.
The object of the article is the first price sealed bid auction consisting of two agents, and the subject of the article are the problems of design and implementation of the first price sealed bid auction.
Methodology
First price sealed bid auction has been analyzed in the article. During first price sealed bid auction the participants of the auction make their bids simultaneously. The product is sold to the agent, who made the highest bid, and the agents pays the bided amount. [4, pp. 421-443], [5, pp. 888-907]. First price sealed bid auction with one indivisible good is considered in the article.
Within the mechanism design theory the term 9i represents the type of ith agent. With the use of an economic mechanism several agents make a collective decision, but prior to decision-making each agent privately observes a distinct parameter or
(1) r = (S1, S2.....SN, g(•))
(2) siESiER+
(3) g(b) = xeX =
Mechanism r = (Si, S2,..., SN, g(•))
implements the social choice function, if there exists a strategy vector (s^,which results in equilibrium state for the mechanism r (Nash equilibrium, Bayesian Nash equilibrium, etc.) [7,
"message", which delineates his preferences and, consequently influences his utility function. Mathematically, this concept is articulated by incorporating the parameter 0;, which is exclusively observed by agent.
The inclusion of 9t in the utility function Ui(a, 0(), signifies the idea, that an agent's type directly influences his preferences and utility function. In case of first price sealed bid auction the type 9i indicates the willingness to pay of the agent for the product, while the 0( is the set of all possible types [6, pp. 857-897].
/: X02 X ... social choice
function defines an alternative f(9) = x e X for each type vector 9 = (9t, 92, d3,...,dN). Social choice function is ex post efficient or Pareto efficient, if for any type vector 9 = (0!, 92,93,..., 9n) there is no alternative xe X, where u^x, 9i)>ui{f{9), 9¿) for each agent i, and U;(x, 9i) >Ui(f(9), 9i) for some agent. The social choice function of the first price sealed bid auction is Pareto efficient, if the product is sold to the agent with the highest valuation.
Mechanism r =(5!,S2,...,SN,g(•)) is a set of N strategy sets (Sx,S2,...,SN) and g: xS2 x •••xSN^X decision function: Each agent i observes his 9t type and based on St strategy sends a message to the mechanism, which makes a collective decision based on gQ) decision function and chooses an alternative x from the set of alternatives X. In first price sealed bid auction all agents privately observe their own types and make bids based on some strategy. According to the decision function the product is sold to the agent, who places the highest bid(yi(ft) = 1 if = max{b1, b2, •••, bN},0' otherwise), where bi is the bid placed by the ith agent, and b is the vector of placed bids.
In case of an auction the alternative also consists of t( payments, which align the decision function of the mechanism with the highest placed bid t( = —bi xyf(A): The mathematical description of the auction as a mechanism is given bellow:
pp. 75-313], where the decision function and the social choice function are equal
S*2.2).....S*n(9n)) =
/(01, 02, 03,..-,9n).
yt{b) = 1 hphbi = max{b1, b2, •••, bN] yt(b) = 0 hphbi ^ max{b1, b2, •••, bN] ti = -bixyi(b)
In essence first price sealed bid auction is a direct mechanism, as the agents provide information regarding their types (willingness to pay). The social choice function implemented by first price sealed bid auction is given in the analysis part of the article.
Social choice function f is truthfully implementable (incentive compatible) [8, pp. 7693], if the direct mechanism r =(0i, 02,—,@n,/(•)) there exists a vector of strategies (s^(91), s2(92),.,s^(9N)) which results in an equilibrium state, where s*(0() = 9^ for all 9i E&i and i. Therefore, the social choice function is truthfully implementable, if the strategy of thrutful revelation of types results in an equilibrium state of the mechanism r = (0i, 02,/(0). The evaluation of truthful implementation or incentive compatibility of the first price sealed bid auction is given in the analysis part of the article.
In case of first price sealed bid auction the strategies of the agents depend on expectations of the strategies of other agents, therefore the solution to the problem (the vector of equilibrium strategies) cannot be dominant strategy Nash equilibrium, but a Bayesian Nash equilibrium. [9, pp. 85-102]. In case of first price sealed bid auction rational agents try to maximize the expected utility, and as the expected utility is highest in case of Bayesian Nash equilibrium strategies, then this is the strategy to be implemented. The vector of strategies (s^(91),s2(92),.,s^(9N)) is considered to be Bayesian Nash equilibrium strategy for the mechanism r = (Si,S2,...,SN, $(•)), if for all agents i and all types 91 the expected utility resulting from implementation of Bayesian Nash equilibrium strategy is higher than the expected utility resulting from any other strategies [10, pp. 296-334].
(4) Ee_i[ui(g(s^(9i),sU), 0i)\9i]> Ee_i[ui(g(s'i(9i),s*_i), 0i)|0i]
for V i, 9t, s', S-i
Mechanism r =(5!, S2,..., SN, g(•))
implements social choice function f with a Bayesian Nash equilibrium strategy, if for the mechanism r there exists Bayesian Nash equilibrium strategy vector
(s^(91),s2(92),...,Sn(9n)), which results to an equilibrium state for the mechanism, where the decision function is equal to the social choice function g(s*(9))=f(9), for all 0 6 0. If there exist a Bayesian Nash equilibrium strategy for the first price sealed bid auction, then it implements some social choice function.
The social choice function is truthfully implementable with Bayesian Nash equilibrium strategy (Bayesian Nash incentive compatible), if the strategy s*(9) = 9 is Bayesian Nash equilibrium strategy for the mechanism
r =(©!, 02,...,0N,/(•)). According to the revelation principle, if there exists a mechanism r = {S1, S2,... ,SN, g(.•)), which implements a social choice function f in Bayesian Nash equilibrium, then the social choice function f is truthfully implementable in Bayesian Nash equilibrium.
Analysis
In case of first price sealed bid auction the buyers place their bids simultaneously. The product is sold to the agent, who placed the highest bid, and the agent pays his bid. In the scope of the analysis first price sealed bid auction with one indivisible good is considered.
The set of / = {1,2} agents is considered. The set of mutually exclusive alternatives X is considered x6l The set of alternatives for the first price sealed bid auction is given bellow:
(5) X = {{y1, y2 , t2): Ji = {0,1} k ti6Rpn^p i hwrfwp, ^ Ji = 1, Ei k < 0) }
The alternative x consists of yt variable, which is 1 or 0 for each agent i. If yi = 1, then the product is bought by ith agent, and if yi = 0, the product is not sold to ith agent: The condition EiJi = 1 ensures, that the product is sold to only one agent. The alternative x also consists of t( variable, which shows the payment of ith agent (the payments have negative sign: Z ti ^ 0): Therefore, in case of first price sealed bid auction the alternative x shows to whom the product is sold and how much money is paid by each agent [11, pp. 1-30].
The payments t( for first price sealed bid auction are given in the formula bellow:
(6) ti = biyi,
where the bi is the placed bid by the agent. If the agent wins the auction, then he pays his bid yt = 1 and ti = bix 1 = bi. Meanwile, if the agent does not win the auction, he does not pay anything yi = 0, ^ub ti=biX 0 = 0:
The utility function u^a, 9¿) of the agent depends on 9t parameter and the result a of the auction (to whom the product is sold and how much is paid for the product). In case of first price sealed
bid auction the type 9t shows the willingness to pay of the agent. The utility function of the agents in the first price sealed bid auction is given bellow:
(7) Ui(a, di) = diyi -ti = 6iyi -btfi
The type of ith agent is represented by 9t, and the set of possible types by 0(. 0 is the vector of types of all agents 0 = (01; 92,03,..,0W). It is
assumed that types 9t are random variables, which have a uniform distribution [12, pp. 1-85], [13, pp. 176-182]: 9i types are normalized within the range of [0,1]. The probability distribution function and cumulative distribution function of uniform distribution are given bellow:
— = — = 0.04, if 0 <9;< 1
b-a 1-0 ' ' 1
0, if 1 >9t or 1 <9i 0, if 1 >9i
0i-O
1 Si
, if 0 <9t< 1 1, if 1 <0t
First price sealed bid auction is represented by the mechanism r = (Si,S2, gO). As the set of agents consist of two agents, the set of strategies also consist of two objects
The ith agent observes his type 9t (known only to him), and according to bi function places a bid b.
The objective of the ith agent in first price sealed bid auction is to maximize his expected utility by choosing a strategy function bt:
(10) max{{9i-bi{9i))F{bi{9i))
(11) F(bi(6i) = Prob(bi>b-i)
(12) max((01-61(01))F(61(01))
(13) max((02-62(02))F(62(02))
The ith agent chooses such a bid function bt, which maximises the product of his utility and probability of winning. In this case the problem of the agent can be solved by first order differentiation.
(14) max((0( ~bi(9i))F(bi(9i)) = max ((^ -M^))^^)
(15) (9ibi(9i) -bi(9i)bi(9i))' = {9ibi{9i) -bi{9i)2)' = 8i- 2bi{9i) = 0
(16) di- 2bi(8i) = 0
According to the analysis above, it can be
stated, that the optimal strategy of the agent, which
0 '
maximizes his expected utility is the strategy -j.
Therefore, according to the optimal strategy the agent having 91 type places a bid which is the half of his type. As the abovementioned strategy is optimal strategy for both of the agents, the
implementation of the optimal strategies results in equilibrium state. Meanwhile, by the description of the equilibrium state the social choice function implemented by the mechanism of the auction can be revealed. The implementation of Bayesian Nash equilibrium strategies results in the following equilibrium state.
(17) 3^(0) = 1 bpb 9i >02 ; yi(0) = 0 hph 9i <02 y2(0) = 1 bpb 02 >02 ; yx(0) = 0 bpb 02 <0! t1(9) = -^r01y1(0) ; t2(0) = -L 92y2(9)
The mechanism r =(5!, S2,...,SN, g(•)) implements the social choice function f with Bayesian Nash equilibrium strategies, if there exists Bayesian Nash equilibrium strategy vector (s^(91), s2(92),...,Sn(9n)) for the mechanism r, which results in such an equilibrium state, where the decision function is equal to the social choice function g(s*(9))=f(9), for all 0 6 0. The 0 '
strategy -j is Bayesian Nash equilibrium strategy, and the results presented in the formula (21) is the
equilibrium state resulting from the implementation of those strategies. Therefore, it can be stated, that first price sealed bid auction implements the social choice function which is presented in the formula (21).
It is obvious, that first price sealed bid auction
is not incentive compatible. Particularly, the agent
with 0; type does not reveal his true type 9t, but
0 '
prefers to reveal another -j type, from which the expected utility is higher. According to incentive compatibility constraint the truthful revelation of
types is optimal strategy, which is not true for the first price sealed bid auction.
According to the revelation principle, if there exists a mechanism r =(5!, S2,...,SN, gO), which implements the social choice function f in Bayesian Nash equilibrium, then the social choice function f is truthfully implementable in Bayesian Nash equilibrium. First price sealed bid auction implements the social choice function represented in the formula (21) in Bayesian Nash equilibrium, therefore, according to revelation principle, there exists another direct mechanism, which truthfully
implements the social choice function represented in the formula (21).
The mechanism r = (Si, S2, gO) is considered. The decision function gQ) is again yi = {0,1}. If yt = 1, then the product is bought by ith agent as bt >bQ, and if yi = 0, then the product is not sold to ith agent as bt <b0. In order to make the mechanism truthfully implementable the payment function should be adjusted, more particularly instead of ti = biyi, transfer function is
adjusted to t( = The objective of the agent in the new mechanism is the following.
(18) max((0,-^)F(W)) = max ((0, ^f^)
(19) vMed - ^ito))' = {eMeo - =Qi-2-^ = 0
(20) 9i-bi(9i) = o obi(ei) = ei
It is obvious from the analysis above, that the adjusted mechanism truthfully implements the social choice function represented in the formula (21) with Bayesian Nash equilibrium strategy. In this case, according to the Bayesian Nash
equilibrium strategy the agents having type 0t place
0 '
bids not equal to -j, but reveal their true 0£ types instead. The designed mechanism is truthfully implementable and incentive compatible.
According to the objectives of the article the expected utilities of the agents and expected revenue
of the seller should be calculated. As mentioned above, the types 0t of the agents are random
variables uniformly distributed in the range of [0,1].
0 '
In case of having type 0£ the ith agent places a bid -j and in case of winning that amount.
As the types are random variable with uniform distribution, then the expected utilities of agents can be calculated by integrating the utility function and the cumulative distribution function in the range of [0,1].
(21) -^F^dx = =
(22) £(^-^ = «¿-«¿=1-1 = 1
J0 \ 1 2 / 3 6 366
Based on formula (26) it can be deduced, that, expected utility of the ith agent from the first price if the types of the agents 0t are random variables sealed bid auction is 1/6.
with uniform distribution in the range of [0,1], the The seller expects to receive from the winning
agent an amount -j:
(23) s = -(t!(0) + t2(0))= \eiyi{9)+\d2y2{9)
As the types of the agents are random variables probability of winning in the range of [0,1]. And the with uniform distribution, the expected revenue of expected revenue from the overall auction can be the seller can be calculated by integrating the calculated by summing expected revenue from each product of payment function £¿(0) and the agent.
(24) s = /0L i1(01)F(01)dx + s = ;01 t2(02)F(02)dx
(25) s = /O1^01F(01) + Jo1^02F(02) dx = J^2 dx + tfjd,2 dx
(26) /n1l012 = 1^ = 1 jn1l02^=il! = I
vy ■'0 21 23 6 ■'0 2z 23 6
f-i'7^ _ 1 i 1 _ 1
v J 6 6 3
From the formula (31 it can be deduced, that, if the types of the agents are random variables uniformly distributed in the range of [0,1], the expected revenue of the seller from each agent is 1/6, and the overall expected revenue from auction is 1/3:
The conclusions derived regarding first price sealed bid auction consisting of two agents are summarized in the Table 1.
Table 1. Conclusions regarding first price sealed bid auction
First price sealed bid auction
Number of agents 2
Bayesian Nash equilibrium 2
Equilibrium state ^(0) = obphtfi < e2 y2(e) = l hpb e2 > e2 yi(0) = ohpte2 < e1 tl(8) = -^e1y1(e)
Implemented social choice function y1(0) - 0 tsph o1 < e2 y2( (?) = ikphe2>e2 y1(e) - o hph e2 < Si tiCfl) = -\eiyi{9) t2(e) = -^e2y2(o)
The mechanism which truthfully implements the social choice function 1 2
Probability of winning of an agent
Expected utility of an agent 1/6 = 0.1667
Expected revenue of the seller 1/3 = 0.3333
Conclusions and recommendations
As a result of the analysis the following conclusions are derived:
• In first price sealed bid auction consisting of
two agents the optimal or Bayesian Nash
equilibrium strategy, which maximizes the expected
0 *
utility of an agent is the strategy -j.
• The Bayesian Nash equilibrium state of the mechanism, which describes first price sealed bid auction is represented in the formula (21), which also implements the social choice function.
• First price sealed bid auction is not truthfully implementable and incentive compatible. In order to truthfully implement the social choice function, a new mechanism is designed by adjusting the payment function = b^yi and replacing it with
ti = function.
1 2
• The probability of winning of an agent in first price sealed bid auction consisting of two agents is di.
• The expected revenue of an agent in first
price sealed bid auction consisting of two agents is
i
6'
• The expected revenue of the seller in first
price sealed bid auction consisting of two agents is
i
3'
Based on the analysis the following recommendation can be made:
• In the article the types are considered as random variables with uniform distribution. It is recommended to consider other statistical distribution and to observe their influence on the results of the model.
• First price sealed bid auction consisting of two agents has been analyzed in the article. It is recommended to give the general solution of the auction with N agents. It is also recommended to evaluate the effect of increase of the number of agents on the results of the auction.
• To make an empirical analysis based on statistical surveys or scientific experiments, giving an empirical assessment of the predictions made in the article.
• The analysis and resulting conclusions can be used to predict the results of first price sealed bid auction consisting of two agents.
Literature
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