Bid Compensation Decision Model for Projectswith Costly Bid PreparationS.Ping Ho,A.M.ASCE 1Abstract:For projects with high bid preparation cost,it is often suggested that the owner should consider paying bid compensation to the most highly ranked unsuccessful bidders to stimulate extra effort or inputs in bid preparation.Whereas the underlying idea of using bid compensation is intuitively sound,there is no theoretical basis or empirical evidence for such suggestion.Because costly bid preparation often implies a larger project scale,the issue of bid compensation strategy is important to practitioners and an interest of study.This paper aims to study the impacts of bid compensation and to develop appropriate bid compensation strategies.Game theory is applied to analyze the behavioral dynamics between competing bidders and project owners.A bid compensation model based on game theoretic analysis is developed in this study.The model provides equilibrium solutions under bid compensation,quantitative formula,and quali-tative implications for the formation of bid compensation strategies.DOI:10.1061/(ASCE )0733-9364(2005)131:2(151)CE Database subject headings:Bids;Project management;Contracts;Decision making;Design/build;Build/Operate/Transfer;Construction industry .IntroductionAn often seen suggestion in practice for projects with high bid preparation cost is that the owner should consider paying bid compensation,also called a stipend or honorarium,to the unsuc-cessful bidders.For example,according to the Design–build Manual of Practice Document Number 201by Design–Build In-stitute of America (DBIA )(1996a ),it is suggested that that “the owner should consider paying a stipend or honorarium to the unsuccessful proposers”because “excessive submittal require-ments without some compensation is abusive to the design–build industry and discourages quality teams from participating.”In another publication by DBIA (1995),it is also stated that “it is strongly recommended that honorariums be offered to the unsuc-cessful proposers”and that “the provision of reasonable compen-sation will encourage the more sought-after design–build teams to apply and,if short listed,to make an extra effort in the prepara-tion of their proposal.”Whereas bid preparation costs depend on project scale,delivery method,and other factors,the cost of pre-paring a proposal is often relatively high in some particular project delivery schemes,such as design–build or build–operate–transfer (BOT )contracting.Plus,costly bid preparation often im-plying a large project scale,the issue of bid compensation strat-egy should be important to practitioners and of great interest of study.Existing research on the procurement process in constructionhas addressed the selection of projects that are appropriate for certain project delivery methods (Molenaar and Songer 1998;Molenaar and Gransberg 2001),the design–build project procure-ment processes (Songer et al.1994;Gransberg and Senadheera 1999;Palaneeswaran and Kumaraswamy 2000),and the BOT project procurement process (United Nations Industrial Develop-ment Organization 1996).However,the bid compensation strat-egy for projects with a relatively high bid preparation cost has not been studied.Among the issues over the bidder’s response to the owner’s procurement or bid compensation strategy,it is in own-er’s interest to understand how the owner can stimulate high-quality inputs or extra effort from the bidder during bid prepara-tion.Whereas the argument for using bid compensation is intuitively sound,there is no theoretical basis or empirical evi-dence for such an argument.Therefore,it is crucial to study under what conditions the bid compensation is effective,and how much compensation is adequate with respect to different bidding situa-tions.This paper focuses on theoretically studying the impacts of bid compensation and tries to develop appropriate compensation strategies for projects with a costly bid preparation.Game theory will be applied to analyze the behavioral dynamics between com-peting bidders.Based on the game theoretic analysis and numeric trials,a bid compensation model is developed.The model pro-vides a quantitative framework,as well as qualitative implica-tions,on bid compensation strategies.Research Methodology:Game TheoryGame theory can be defined as “the study of mathematical models of conflict and cooperation between intelligent rational decision-makers”(Myerson 1991).Among economic theories,game theory has been successfully applied to many important issues such as negotiations,finance,and imperfect markets.Game theory has also been applied to construction management in two areas.Ho (2001)applied game theory to analyze the information asymme-try problem during the procurement of a BOT project and its1Assistant Professor,Dept.of Civil Engineering,National Taiwan Univ.,Taipei 10617,Taiwan.E-mail:spingho@.twNote.Discussion open until July 1,2005.Separate discussions must be submitted for individual papers.To extend the closing date by one month,a written request must be filed with the ASCE Managing Editor.The manuscript for this paper was submitted for review and possible publication on March 5,2003;approved on March 1,2004.This paper is part of the Journal of Construction Engineering and Management ,V ol.131,No.2,February 1,2005.©ASCE,ISSN 0733-9364/2005/2-151–159/$25.00.D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y N A N J I N G U N I VE R S I T Y OF o n 01/06/14. C o p y r i g h t A S C E . F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .implication in project financing and government policy.Ho and Liu (2004)develop a game theoretic model for analyzing the behavioral dynamics of builders and owners in construction claims.In competitive bidding,the strategic interactions among competing bidders and that between bidders and owners are com-mon,and thus game theory is a natural tool to analyze the prob-lem of concern.A well-known example of a game is the “prisoner’s dilemma”shown in Fig.1.Two suspects are arrested and held in separate cells.If both of them confess,then they will be sentenced to jail for 6years.If neither confesses,each will be sentenced for only 1year.However,if one of them confesses and the other does not,then the honest one will be rewarded by being released (in jail for 0year )and the other will be punished for 9years in jail.Note that in each cell,the first number represents player No.1’s payoff and the second one represents player No.2’s.The prisoner’s dilemma is called a “static game,”in which they act simultaneously;i.e.,each player does not know the other player’s decision before the player makes the decision.If the payoff matrix shown in Fig.1is known to all players,then the payoff matrix is a “common knowledge”to all players and this game is called a game of “complete information.”Note that the players of a game are assumed to be rational;i.e.,to maximize their payoffs.To answer what each prisoner will play/behave in this game,we will introduce the concept of “Nash equilibrium ,”one of the most important concepts in game theory.Nash equilibrium is a set of actions that will be chosen by each player.In a Nash equilib-rium,each player’s strategy should be the best response to the other player’s strategy,and no player wants to deviate from the equilibrium solution.Thus,the equilibrium or solution is “strate-gically stable”or “self-enforcing”(Gibbons 1992).Conversely,a nonequilibrium solution is not stable since at least one of the players can be better off by deviating from the nonequilibrium solution.In the prisoner’s dilemma,only the (confess,confess )solution where both players choose to confess,satisfies the stabil-ity test or requirement of Nash equilibrium.Note that although the (not confess,not confess )solution seems better off for both players compared to Nash equilibrium;however,this solution is unstable since either player can obtain extra benefit by deviating from this solution.Interested readers can refer to Gibbons (1992),Fudenberg and Tirole (1992),and Myerson (1991).Bid Compensation ModelIn this section,the bid compensation model is developed on the basis of game theoretic analysis.The model could help the ownerform bid compensation strategies under various competition situ-ations and project characteristics.Illustrative examples with nu-merical results are given when necessary to show how the model can be used in various scenarios.Assumptions and Model SetupTo perform a game theoretic study,it is critical to make necessary simplifications so that one can focus on the issues of concern and obtain insightful results.Then,the setup of a model will follow.The assumptions made in this model are summarized as follows.Note that these assumptions can be relaxed in future studies for more general purposes.1.Average bidders:The bidders are equally good,in terms oftheir technical and managerial capabilities.Since the design–build and BOT focus on quality issues,the prequalification process imposed during procurement reduces the variation of the quality of bidders.As a result,it is not unreasonable to make the “average bidders”assumption.plete information:If all players consider each other tobe an average bidder as suggested in the first assumption,it is natural to assume that the payoffs of each player in each potential solution are known to all players.3.Bid compensation for the second best bidder:Since DBIA’s(1996b )manual,document number 103,suggests that “the stipend is paid only to the most highly ranked unsuccessful offerors to prevent proposals being submitted simply to ob-tain a stipend,”we shall assume that the bid compensation will be offered to the second best bidder.4.Two levels of efforts:It is assumed that there are two levelsof efforts in preparing a proposal,high and average,denoted by H and A ,respectively.The effort A is defined as the level of effort that does not incur extra cost to improve quality.Contrarily,the effort H is defined as the level of effort that will incur extra cost,denoted as E ,to improve the quality of a proposal,where the improvement is detectable by an effec-tive proposal evaluation system.Typically,the standard of quality would be transformed to the evaluation criteria and their respective weights specified in the Request for Pro-posal.5.Fixed amount of bid compensation,S :The fixed amount canbe expressed by a certain percentage of the average profit,denoted as P ,assumed during the procurement by an average bidder.6.Absorption of extra cost,E :For convenience,it is assumedthat E will not be included in the bid price so that the high effort bidder will win the contract under the price–quality competition,such as best-value approach.This assumption simplifies the tradeoff between quality improvement and bid price increase.Two-Bidder GameIn this game,there are only two qualified bidders.The possible payoffs for each bidder in the game are shown in a normal form in Fig.2.If both bidders choose “H ,”denoted by ͑H ,H ͒,both bidders will have a 50%probability of wining the contract,and at the same time,have another 50%probability of losing the con-tract but being rewarded with the bid compensation,S .As a re-sult,the expected payoffs for the bidders in ͑H ,H ͒solution are ͑S /2+P /2−E ,S /2+P /2−E ͒.Note that the computation of the expected payoff is based on the assumption of the average bidder.Similarly,if the bidders choose ͑A ,A ͒,the expected payoffswillFig.1.Prisoner’s dilemmaD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y N A N J I N G U N I VE R S I T Y OF o n 01/06/14. C o p y r i g h t A S C E . F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .be ͑S /2+P /2,S /2+P /2͒.If the bidders choose ͑H ,A ͒,bidder No.1will have a 100%probability of winning the contract,and thus the expected payoffs are ͑P −E ,S ͒.Similarly,if the bidders choose ͑A ,H ͒,the expected payoffs will be ͑S ,P −E ͒.Payoffs of an n -bidder game can be obtained by the same reasoning.Nash EquilibriumSince the payoffs in each equilibrium are expressed as functions of S ,P ,and E ,instead of a particular number,the model will focus on the conditions for each possible Nash equilibrium of the game.Here,the approach to solving for Nash equilibrium is to find conditions that ensure the stability or self-enforcing require-ment of Nash equilibrium.This technique will be applied throughout this paper.First,check the payoffs of ͑H ,H ͒solution.For bidder No.1or 2not to deviate from this solution,we must haveS /2+P /2−E ϾS →S ϽP −2E͑1͒Therefore,condition (1)guarantees ͑H ,H ͒to be a Nash equilib-rium.Second,check the payoffs of ͑A ,A ͒solution.For bidder No.1or 2not to deviate from ͑A ,A ͒,condition (2)must be satisfiedS /2+P /2ϾP −E →S ϾP −2E͑2͒Thus,condition (2)guarantees ͑A ,A ͒to be a Nash equilibrium.Note that the condition “S =P −2E ”will be ignored since the con-dition can become (1)or (2)by adding or subtracting an infinitely small positive number.Thus,since S must satisfy either condition (1)or condition (2),either ͑H ,H ͒or ͑A ,A ͒must be a unique Nash equilibrium.Third,check the payoffs of ͑H ,A ͒solution.For bid-der No.1not to deviate from H to A ,we must have P −E ϾS /2+P /2;i.e.,S ϽP −2E .For bidder No.2not to deviate from A to H ,we must have S ϾS /2+P /2−E ;i.e.,S ϾP −2E .Since S cannot be greater than and less than P −2E at the same time,͑H ,A ͒solution cannot exist.Similarly,͑A ,H ͒solution cannot exist either.This also confirms the previous conclusion that either ͑H ,H ͒or ͑A ,A ͒must be a unique Nash equilibrium.Impacts of Bid CompensationBid compensation is designed to serve as an incentive to induce bidders to make high effort.Therefore,the concerns of bid com-pensation strategy should focus on whether S can induce high effort and how effective it is.According to the equilibrium solu-tions,the bid compensation decision should depend on the mag-nitude of P −2E or the relative magnitude of E compared to P .If E is relatively small such that P Ͼ2E ,then P −2E will be positive and condition (1)will be satisfied even when S =0.This means that bid compensation is not an incentive for high effort when the extra cost of high effort is relatively low.Moreover,surprisingly,S can be damaging when S is high enough such that S ϾP −2E .On the other hand,if E is relatively large so that P −2E is negative,then condition (2)will always be satisfied since S can-not be negative.In this case,͑A ,A ͒will be a unique Nash equi-librium.In other words,when E is relatively large,it is not in the bidder’s interest to incur extra cost for improving the quality of proposal,and therefore,S cannot provide any incentives for high effort.To summarize,when E is relatively low,it is in the bidder’s interest to make high effort even if there is no bid compensation.When E is relatively high,the bidder will be better off by making average effort.In other words,bid compensation cannot promote extra effort in a two-bidder game,and ironically,bid compensa-tion may discourage high effort if the compensation is too much.Thus,in the two-bidder procurement,the owner should not use bid compensation as an incentive to induce high effort.Three-Bidder GameNash EquilibriumFig.3shows all the combinations of actions and their respective payoffs in a three-bidder game.Similar to the two-bidder game,here the Nash equilibrium can be solved by ensuring the stability of the solution.For equilibrium ͑H ,H ,H ͒,condition (3)must be satisfied for stability requirementS /3+P /3−E Ͼ0→S Ͼ3E −P͑3͒For equilibrium ͑A ,A ,A ͒,condition (4)must be satisfied so that no one has any incentives to choose HS /3+P /3ϾP −E →S Ͼ2P −3E͑4͒In a three-bidder game,it is possible that S will satisfy conditions (3)and (4)at the same time.This is different from the two-bidder game,where S can only satisfy either condition (1)or (2).Thus,there will be two pure strategy Nash equilibria when S satisfies conditions (3)and (4).However,since the payoff of ͑A ,A ,A ͒,S /3+P /3,is greater than the payoff of ͑H ,H ,H ͒,S /3+P /3−E ,for all bidders,the bidder will choose ͑A ,A ,A ͒eventually,pro-vided that a consensus between bidders of making effort A can be reached.The process of reaching such consensus is called “cheap talk,”where the agreement is beneficial to all players,and no player will want to deviate from such an agreement.In the design–build or BOT procurement,it is reasonable to believe that cheap talk can occur.Therefore,as long as condition (4)is satis-fied,͑A ,A ,A ͒will be a unique Nash equilibrium.An important implication is that the cheap talk condition must not be satisfied for any equilibrium solution other than ͑A ,A ,A ͒.In other words,condition (5)must be satisfied for all equilibrium solution except ͑A ,A ,A͒Fig.2.Two-biddergameFig.3.Three-bidder gameD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y N A N J I N G U N I VE R S I T Y OF o n 01/06/14. C o p y r i g h t A S C E . F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .S Ͻ2P −3E ͑5͒Following this result,for ͑H ,H ,H ͒to be unique,conditions (3)and (5)must be satisfied;i.e.,we must have3E −P ϽS Ͻ2P −3E͑6͒Note that by definition S is a non-negative number;thus,if one cannot find a non-negative number to satisfy the equilibrium con-dition,then the respective equilibrium does not exist and the equi-librium condition will be marked as “N/A”in the illustrative fig-ures and tables.Next,check the solution where two bidders make high efforts and one bidder makes average effort,e.g.,͑H ,H ,A ͒.The ex-pected payoffs for ͑H ,H ,A ͒are ͑S /2+P /2−E ,S /2+P /2−E ,0͒.For ͑H ,H ,A ͒to be a Nash equilibrium,S /3+P /3−E Ͻ0must be satisfied so that the bidder with average effort will not deviate from A to H ,S /2+P /2−E ϾS /2must be satisfied so that the bidder with high effort will not deviate from H to A ,and condi-tion (5)must be satisfied as argued previously.The three condi-tions can be rewritten asS Ͻmin ͓3E −P ,2P −3E ͔andP −2E Ͼ0͑7͒Note that because of the average bidder assumption,if ͑H ,H ,A ͒is a Nash equilibrium,then ͑H ,A ,H ͒and ͑A ,H ,H ͒will also be the Nash equilibria.The three Nash equilibria will constitute a so-called mixed strategy Nash equilibrium,denoted by 2H +1A ,where each bidder randomizes actions between H and A with certain probabilities.The concept of mixed strategy Nash equilib-rium shall be explained in more detail in next section.Similarly,we can obtain the requirements for solution 1H +2A ,condition (5)and S /2+P /2−E ϽS /2must be satisfied.The requirements can be reorganized asS Ͻ2P −3EandP −2E Ͻ0͑8͒Note that the conflicting relationship between “P −2E Ͼ0”in condition (7)and “P −2E Ͻ0”in condition (8)seems to show that the two types of Nash equilibria are exclusive.Nevertheless,the only difference between 2H +1A and 1H +2A is that the bidder in 2H +1A equilibrium has a higher probability of playing H ,whereas the bidder in 1H +2A also mixes actions H and A but with lower probability of playing H .From this perspective,the difference between 2H +1A and 1H +2A is not very distinctive.In other words,one should not consider,for example,2H +1A ,to be two bidders playing H and one bidder playing A ;instead,one should consider each bidder to be playing H with higher probabil-ity.Similarly,1H +2A means that the bidder has a lower probabil-ity of playing H ,compared to 2H +1A .Illustrative Example:Effectiveness of Bid Compensation The equilibrium conditions for a three-bidder game is numerically illustrated and shown in Table 1,where P is arbitrarily assumed as 10%for numerical computation purposes and E varies to rep-resent different costs for higher efforts.The “*”in Table 1indi-cates that the zero compensation is the best strategy;i.e.,bid compensation is ineffective in terms of stimulating extra effort.According to the numerical results,Table 1shows that bid com-pensation can promote higher effort only when E is within the range of P /3ϽE ϽP /2,where zero compensation is not neces-sarily the best strategy.The question is that whether it is benefi-cial to the owner by incurring the cost of bid compensation when P /3ϽE ϽP /2.The answer to this question lies in the concept and definition of the mix strategy Nash equilibrium,2H +1A ,as explained previously.Since 2H +1A indicates that each bidderwill play H with significantly higher probability,2H +1A may already be good enough,knowing that we only need one bidder out of three to actually play H .We shall elaborate on this concept later in a more general setting.As a result,if the 2H +1A equilib-rium is good enough,the use of bid compensation in a three-bidder game will not be recommended.Four-Bidder Game and n-Bidder GameNash Equilibrium of Four-Bidder GameThe equilibrium of the four-bidder procurement can also be ob-tained.As the number of bidders increases,the number of poten-tial equilibria increases as well.Due to the length limitation,we shall only show the major equilibria and their conditions,which are derived following the same technique applied previously.The condition for pure strategy equilibrium 4H ,is4E −P ϽS Ͻ3P −4E͑9͒The condition for another pure strategy equilibrium,4A ,isS Ͼ3P −4E͑10͒Other potential equilibria are mainly mixed strategies,such as 3H +1A ,2H +2A ,and 1H +3A ,where the numeric number asso-ciated with H or A represents the number of bidders with effort H or A in a equilibrium.The condition for the 3H +1A equilibrium is3E −P ϽS Ͻmin ͓4E −P ,3P −4E ͔͑11͒For the 2H +2A equilibrium the condition is6E −3P ϽS Ͻmin ͓3E −P ,3P −4E ͔͑12͒The condition for the 1H +3A equilibrium isS Ͻmin ͓6E −3P ,3P −4E ͔͑13͒Illustrative Example of Four-Bidder GameTable 2numerically illustrates the impacts of bid compensation on the four-bidder procurement under different relative magni-tudes of E .When E is very small,bid compensation is not needed for promoting effort H .However,when E grows gradually,bid compensation becomes more effective.As E grows to a larger magnitude,greater than P /2,the 4H equilibrium would become impossible,no matter how large S is.In fact,if S is too large,bidders will be encouraged to take effort A .When E is extremely large,e.g.,E Ͼ0.6P ,the best strategy is to set S =0.The “*”in Table 2also indicates the cases that bid compensation is ineffec-Table pensation Impacts on a Three-Bidder GameEquilibriumE ;P =10%3H 2H +1A 1H +2A 3A E ϽP /3e.g.,E =2%S Ͻ14%*N/A N/N 14%ϽS P /3ϽE ϽP /2e.g.,E =4%2%ϽS Ͻ8%S Ͻ2%N/A 8%ϽS P /2ϽE Ͻ͑2/3͒P e.g.,E =5.5%N/AN/AS Ͻ3.5%*3.5%ϽS͑2/3͒P ϽEe.g.,E =7%N/A N/A N/A Always*Note:*denotes that zero compensation is the best strategy;and N/A =the respective equilibrium does not exist.D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y N A N J I N G U N I VE R S I T Y OF o n 01/06/14. C o p y r i g h t A S C E . F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .tive.To conclude,in a four-bidder procurement,bid compensation is not effective when E is relatively small or large.Again,similar to the three-bidder game,when bid compensation becomes more effective,it does not mean that offering bid compensation is the best strategy,since more variables need to be considered.Further analysis shall be performed later.Nash Equilibrium of n -Bidder GameIt is desirable to generalize our model to the n -bidder game,al-though only very limited qualified bidders will be involved in most design–build or BOT procurements,since for other project delivery methods it is possible to have many bidders.Interested readers can follow the numerical illustrations for three-and four-bidder games to obtain the numerical solutions of n -bidder game.Here,only analytical equilibrium solutions will be solved.For “nA ”to be the Nash equilibrium,we must have P −E ϽS /n +P /n for bidder A not to deviate.In other words,condition (14)must be satisfiedS Ͼ͑n −1͒P −nE͑14͒Note that condition (14)can be rewritten as S Ͼn ͑P −E ͒−P ,which implies that it is not likely for nA to be the Nash equilib-rium when there are many bidders,unless E is very close to or larger than P .Similar to previous analysis,for “nH ”to be the equilibrium,we must have S /n +P /n −E Ͼ0for stability requirement,and condition (15)for excluding the possibility of cheap talk or nA equilibrium.The condition for the nH equilibrium can be reorga-nized as condition (16).S Ͻ͑n −1͒P −nE ͑15͒nE −P ϽS Ͻ͑n −1͒P −nE͑16͒Note that if E ϽP /n ,condition (16)will always be satisfied and nH will be a unique equilibrium even when S =0.In other words,nH will not be the Nash equilibrium when there are many bidders,unless E is extremely small,i.e.,E ϽP /n .For “aH +͑n −a ͒A ,where 2Ͻa Ͻn ”to be the equilibrium so-lution,we must have S /a +P /a −E Ͼ0for bidder H not to devi-ate,S /͑a +1͒+P /͑a +1͒−E Ͻ0for bidder A not to deviate,and condition (15).These requirements can be rewritten asaE −P ϽS Ͻmin ͓͑a +1͒E −P ,͑n −1͒P −nE ͔͑17͒Similarly,for “2H +͑n −2͒A ,”the stability requirements for bidder H and A are S /͑n −1͒ϽS /2+P /2−E and S /3+P /3−E Ͻ0,re-spectively,and thus the equilibrium condition can be written as ͓͑n −1͒/͑n −3͔͒͑2E −P ͒ϽS Ͻmin ͓3E −P ,͑n −1͒P −nE ͔͑18͒For the “1H +͑n −1͒A ”equilibrium,we must haveS Ͻmin ͕͓͑n −1͒/͑n −3͔͒͑2E −P ͒,͑n −1͒P −nE ͖͑19͒An interesting question is:“What conditions would warrant that the only possible equilibrium of the game is either “1H +͑n −1͒A ”or nA ,no matter how large S is?”A logical response to the question is:when equilibria “aH +͑n −a ͒A ,where a Ͼ2”and equilibrium 2H +͑n −2͒A are not possible solutions.Thus,a suf-ficient condition here is that for any S Ͼ͓͑n −1͒/͑n −3͔͒͑2E −P ͒,the “S Ͻ͑n −1͒P −nE ”is not satisfied.This can be guaranteed if we have͑n −1͒P −nE Ͻ͓͑n −1͒/͑n −3͔͒͑2E −P ͒→E Ͼ͓͑n −1͒/͑n +1͔͒P͑20͒Conditions (19)and (20)show that when E is greater than ͓͑n −1͒/͑n +1͔͒P ,the only possible equilibrium of the game is either 1H +͑n −1͒A or nA ,no matter how large S is.Two important practical implications can be drawn from this finding.First,when n is small in a design–build contract,it is not unusual that E will be greater than ͓͑n −1͒/͑n +1͔͒P ,and in that case,bid compensa-tion cannot help to promote higher effort.For example,for a three-bidder procurement,bid compensation will not be effective when E is greater than ͑2/4͒P .Second,when the number of bidders increases,bid compensation will become more effective since it will be more unlikely that E is greater than ͓͑n −1͒/͑n +1͔͒P .The two implications confirm the previous analyses of two-,three-,and four-bidder game.After the game equilibria and the effective range of bid compensation have been solved,the next important task is to develop the bid compensation strategy with respect to various procurement situations.Table pensation Impacts on a Four-Bidder GameEquilibriumE ;P =10%4H 3H +1A 2H +2A 1H +3A 4A E ϽP /4e.g.,E =2%S Ͻ22%*N/A N/A N/A S Ͼ22%P /4ϽE ϽP /3e.g.,E =3%2%ϽS Ͻ18%S Ͻ2%N/A N/A S Ͼ18%P /3ϽE ϽP /2e.g.,E =4%6%ϽS Ͻ14%2%ϽS Ͻ6%S Ͻ2%N/A S Ͼ14%P /2ϽE Ͻ͑3/5͒P e.g.,E =5.5%N/A 6.5%ϽS Ͻ8%3%ϽS Ͻ6.5%S Ͻ3%S Ͼ8%͑3/5͒P ϽE Ͻ͑3/4͒P e.g.,E =6.5%N/AN/AN/AS Ͻ4%*S Ͼ4%͑3/4͒P ϽEe.g.,E =8%N/A N/A N/A N/AAlways*Note:*denotes that zero compensation is the best strategy;and N/A=respective equilibrium does not exist.D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y N A N J I N G U N I VE R S I T Y OF o n 01/06/14. C o p y r i g h t A S C E . F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .。
