PHAN KIM ANHBID COMPENSATION EFFECTIVENESS ANALYSISBASED ON OWNER AND CONTRACTORPERSPECTIVE USING GAME THEORY APPROACHMajor: CONSTRUCTION MANAGEMENTMajor code: 8580302MASTER’S THESISHO C
INTRODUCTION
Introduction
In a highly competitive environment such as the construction industry, the selection of contractors has long been regarded as one of the most difficult tasks in project management Competitive bidding has been deemed a legal requirement for projects utilizing national budget or loan capital in an effort to combat waste of public funds and prevent abuses, such as fraud, waste, and favoritism, that are the responsibility of state organizations (Rowles, D., and S Cahalan, 2020) 1 The contractor selection element is the most significance of the procurement stage in which low-bid competition has been predominant criteria in the bidding process in worldwide over many years [1] This has pressured on price, proliferation of construction systems and products to fulfill the minimum required specifications, it is easy to cause to construction non-performance and litigation when contractor
1 Rowles, D., and S Cahalan (2020) “Local government procurement laws— Who the heck is a
‘responsible bidder’?” Accessed April 22, 2020 https://www.sgrlaw.com/local-government- procurement-laws-who-the -heck-is-a-responsible-bidder/ faces with producing greater volume Hence, the generated results from low-bid process has not meet the owner's desire “It has produced low-quality work, adversarial working conditions, a high incidence of contractor-generated change orders, claims, litigation and increased project management costs” [2] The abnormally low bids are considered as one of the main causes of poor project quality [3], [4] According to Hwang [5] and Burati [6], design flaws alone are estimated to contribute to 79% of all deviations from the projected cost of a project, and the estimated losses owing to design flaws amount to at least 9.5% of total costs for construction project Thus, such defects may cause financial losses and delay schedules for all stakeholders, especially contractors and the client For the purpose of high performance in projects, the endeavours for initial preparations as planning and design step are played the crucial role [7].
For PPP projects, the requirements for associated project risks, project plan, concept design, regulations during the construction execution, operation, and transfer are requisite at the early stage to implement and perform the project As KPMG (2010) reported, the costs of consultation, design, and market research in the pre-tender stage accounted for an average of 1.5%–2% of total project costs Thus far, these initial costs are barriers for drawing consortias due to uncertainty in awarding the contract [9] Other important obstacles for using PPPs are extended time, high costs for transaction, a lack of competition and transparency, which causes inefficiencies and ineffectiveness in tendering processes [10], [11]. Soomro [12] examined the number of initiating failure mechanisms at each project stage, it is possible to conclude that the feasibility and procurement stages are the most critical for public sector personnel to initiate a variety of failure mechanisms.
As a result, practitioners have sought strategies that encourage contractors to exert additional effort during the early stages of the project's life cycle For example, theDesign–Build Institute of America, or DBIA (1996a), ‘the owner should consider paying a stipend or honorarium to the unsuccessful proposers’ because ‘excessive submittal requirements without some compensation is abusive to the design–build industry and discourages quality teams from participating’ Another publication by the DBIA (2014) stated that ‘owners should offer a reasonable stipend to unsuccessful shortlisted proposers when the proposal preparation requires a significant level of effort’ Although these statements are based on guidelines governing design–build projects, the aforementioned compensation is also considered applicable to other types of projects to prevent illegal practices such as price agreements.
Compensation is viewed as a means of increasing the tender's attractiveness to the private sector and of encouraging bidders to put in additional effort during the preplanning and feasibility study stages Nonetheless, there are few studies on bid compensation in competitive construction bidding Bid compensation is defined as a stipend or honorarium paid by the procuring authority to unsuccessful bidders to cover the additional effort required to prepare a bid [13]–[15] The additional effort outlays may include hiring top-tier advisers and professional consultants, conducting studies that are more thorough than mere preliminary design and analysis, and suggesting optimal alternatives to create value for bidders For example, according to the Mac Leamy Curve as figure 1.1, the efforts in early stage are small level of impact in cost of design changes By expending additional effort and acquiring experience in conducting detailed project analysis, the construction firm should proffer cost and time efficiency alternatives Moreover, from the owner’s perspective, elaborate preparatory analysis may minimize project variations to restrict the contingency expenditure to within the acceptable range In spite of expression of employing bid compensation that is pursuasively sound, its implementation has been limited in circumstances, and it has been neglected the analysis of effectiveness ability in the associated literature; no theoretical framework or set of guidelines has been introduced to assist in the application of bid compensation.
In this study, we used a mathematical method to investigate the characteristics of bid compensation and analyse its effectiveness in developing appropriate owner-oriented strategies for projects for which are expensive costs for bid preparation The theory of game has been implemented to build up the interaction model of owner and bidder and analysed their behaviour Through this game theory model utilization, it has determined a framework of quantifier and the implications of qualifier for the compensation tactics.
Figure 1.1 The Mac Leamy Curve
Research Scope And Objectives
Generally, the project life cycle is divided into several stages such as origination, preparation, procurement, construction, contract management as shown in Figure 1.2 However, to limit the research scope, the study only focuses on one of the most primary issues, the tendering process in procurement stage.
The purpose of this dissertation is to demonstrate the model's forecasting capability by providing assistance and recommendations to both the owner and contractors in determining an appropriate strategy The owner can foresee the contractor’s future bidding strategy to decide the bid compensation amount that compensate worthly for the contractor's effort Moreover, the contractors also can predict the owner's bid compensation strategy to invest more effort in their proposal as well That increases the probability of right contractor selection for the successful project performance and management Also, this research supplements a novel framework in construction management The objectives of the proposed model, as depicted in Figure 1.3, are to:
(1) define the parameters that affect directly the players's strategy,
(2) develop learning approach for different bidding participant condition,
(3) forecasting a high effort strategy of contractor in each scenarios of bid compensation strategy.
Research Methodology
To obtain the objectives, the following methodology and sub-objectives, as seen in Figure 1.4, need to be conducted:
(1) Performing literature review, related to bid compensation model, to find the gap and demerits of previous researches.
(2) Developing a modeling concept of bid compensation model using Game Theory Model.
(3) Examining the case of the homogeneous bidders involving bidding process, verifying and evaluating the proposed model using same case study input developed by Ho (2005).
(4) Examing the case of heterogeneous bidders involving bidding process, comparing among the case of increasing participants.
(5) Analyzing the practicality and acceptability of this study to practice by conducting a bid compensation strategy based on a realistic business environment.
Literature Review of previous studies
Verifying the applicability and practicality
Dissertation Organization
Chapter 1 introduces the importance of bid compensation system to improve effectiveness and efficiency in the construction procurement This chapter also explicitly describes the research scope, research objectives, and research methodology, relevance between the proposed model and management point of view and also this dissertation organization.
Chapter 2 presents general overview of competitive bidding process and potential factors affecting the bid compensation strategy outcomes This chapter provides basic theoretical background about tools used in proposed optimization model such as: competitive game theory model, Nash Equilibrium solution This chapter also provides information about the game theory model and previous studies related to bid compensation The last part of this chapter introduces the proposed models to support discussion in previous study Section.
Chapter 3 presents a detail design and development of bid compensation model for homogeneous bidders It also discusses and verified the results of the proposed model.
Chapter 4 presents the another scenario of proposed model in the form of heterogeneous bidders The objective of this chapter is comparison the impact of bid compensation in increasing the number of players condition
Chapter 5 presents the final conclusion which describes the finding, contribution and recommendation for further research.
In general, the dissertation organization and link among chapters can be seen in the Figure 1.5.
OVERVIEW
General Overview Of Bidding Process In Large-Scale Construction Projects
Large-scale construction projects have typical characteristics such as the mass of capital investments and management uncertainties, lengthy execution period, which make the selection of suitable contractors as a challenging task (Al- Reshaid and Kartam 2005, Hatush and Skitmore 1998, Kashiwagi and Byfield 2002, Watt, Kayis, and Willey 2010, Hatush and Skitmore 1998) Moreover, the large- scale construction projects hold a vital position in the construction industry, as well as, contribute importantly the development of whole national economy-society The success of these projects have the relevance of fixed-asset formation and the stimulation of general national economy growth in future The numerous large-scale construction projects impact to various aspects of citizen as life quality or life security, for example, irrigation facilities, roads, bridges, quarries, etc Thus, their significance is self-evident [19].
2.1.2- The competitive bidding process objectives
The contractor selection is the proceeding of proposal submission to execute, or manage the execution of a construction project The main concern of bidding process is the selection of the right contractor in uncertain environment [20] who may undertake the project on time and on budget.
For large-scale construction projects, the proposals are required high quality- standard for technologies and aesthetic decoration, thus, the construction method and (or) design scheme requirements are more complicated Additionally, the tight construction schedule is to meet the owner's requirement, and the social needs as well Thus, the number of risks and uncertainties increase greatly, and the capability of a contractor to treat risks and uncertainties, which present through spending more effort to study deeply at feasibility stage, becomes critical Especially public projects, the competitive bidding have to be compulsory to prevent political graft and corruption caused by favoritism and fraud The competitive selection of an proper contractor is conducive to economical and timely construction service with the best quality [21] The competitive bidding process involves objectives which are
To ensure the procurement process runs smoothly and without interruptions or rescheduling.
To deliver a contract that is both cost effective and beneficial to both parties.
To conduct the selection process effectively and transparently.
Sufficient qualifications to carry out the project in a manner that maximizes the likelihood of a successful completion.
To effectively leverage competition in order to provide the owner with the best value for money (government).
Generally, the bidding process in most of the construction industry consists of the following steps:
(2) Bid Documents Approval by Competent Authority
(3) Public Invitation for Pre-Qualification
(6) Issuance of Bid Document to prospective bidders
(7) Pre Bid Meeting and Issuance of Minutes, Clarifications, and Common Set of Deviation
(11) Acceptance of Bid and Award Contract
Firstly, the bid document is prepared based on the owner's requisitions and gotten approval from authority According to the characteristics and type form of project, the owner seeks for the potential contractors For those contractors that has experienced in the similar projects, they are usually invited to the bidding process.
In case, other contractors are no experience, the other contractors are needed to meet the criterias through pre-qualification assessment for further steps After prospective bidders listed, the request for bid proposal is a set of writing document that is sent to contractors to be sure competitive and reliable During the bidding process, the contractors have been organized a pre bid meeting, requesting for clarifications and deviations to the owner After the contractors have completed and returned the proposals to the owner, the proposals are analysed for technical and commercial requirements, compliance to specifications, terms and conditions of contract The owner will then conduct negotiation with the selected contractor.
Figure 2.1 The bidding process cycle
2.1.4- The competitive environment of contractor selection process
The method of project delivery contains four main methods that aid the Employers to perform the construction projects within the projected time frame as well as budget The owner awards the contract to the most suitable bidder among all the eligible bidders The generally adopted selection criteria of the most eligible bidder are as shown in figure 2.2.
In many design-build or PPP projects, it is widely assumed that bids are evaluated on the basis of best value, rather than best (lowest) price, and that value is reflected in the bid proposal and interaction between the project owner and bidders.
Figure 2.2 The bidding criteria for project delivery method
2.1.5- Relevant studies in bidding process
Several attempts have been made thus far to propose decision-supporting models for construction bidding The main objective of these studies is to analyze various dimensions of the competitive bidding environment Ngai [22] required the certain number of competitive bidders to satisfy the cost-efficient manner of bidding competition for a projects under varying market condition Lo [4] analyzed collected data of 44 Taiwan highway projects by statistic method, which involves the aspects of cost, competitive characteristic of market and beyond-contractual reward (BCR) to develop the contractor's price model, from which improving the construction management system and opportunistic behavior of contractor selection, reducing the probability of BCR situation gaining through cutting corners and claims as well Wu [23] suggested the function of time which formulates the time-cost relationship to obtain the construction time and cost optimization through the linear regression approach, to aiming at assisting the contractor's strategy deciding in bidding process; in addition, the proposed model is the basis to improve the model of price-time biparameter with more criteria in construction bidding. Leśniak [24] and Al-Humaidi [25] proposed a model of construction bid or not bid to assist the contractor’s making-decision by using fuzzy technique, that presented the most vital factors influencing bid/no bid decision, as well as provided a ranking of different projects to bid on In another study relating to bid or not bid decision model, Li [26] proposed the evidential reasoning (ER) approach to resolve the problem of arising uncertainty from qualitative criteria and incomplete information in construction international projects Ojelabi [27] found out three key potential barrier factors of participating of private investors in PPP construction project bidding process, as political relevance, poor management for PPPs from public sector, and inadequate input of project information Li [28] contributed a novel knowledge of dissecting the deviation of the owner's estimate and the submitted bid of contractor, that identify four risk indicators lead to that difference in highway projects, as well as forecast the ratio of low bid to owner's estimate by the innovatory time-series models developed; hence high accuracy of cost estimate preparation and tactical development of risk management are to enhance decision- making for transportation agencies.
Game theory and Its Application
2.2.1- The introduction of Game Theory
Game theory is defined as "the study of mathematical models of conflict and cooperation between rational intelligent agents" (Myerson, 1991) Von Neumann and Morgenstern published “The Theory of Games and Economic Behavior” in
1944, that is the first time to introduce the definition of game theory (Resmusen,2005) The next edition of this publication assisted mathematical statisticians and economists in making decision under uncertainty by an axiomatic theory of expected utility providing In the beginning, zero-sum game method was addressed, in which each participant are precise balancing of gains or losses by those of others. From the 1950s to the 1970s, game theory experienced a surge of activity and won numerous Nobel prizes, during which a flurry of definitions were developed, including the core, fictitious play, repeated games, the extensive form game, and the Shapley value Nowadays, it has applications in a wide variety of fields, including social science, logic, systems science, and computer science [29]–[35].
The essential elements of a game are players, actions, payoffs, and information.
Individuals who make decisions are referred to as players Each player's objective is to maximize his utility through action selection.
An action or a move represents a decision he can make.
Payoff refers to the utility received by a player after all other players and Nature have chosen their strategies and the game has been completed; or the expected utility received as a function of his and the other players' strategies.
A player's information set is defined as his current knowledge of the values of various variables.
In 1951, John Nash introduced non-cooperative game theory in his article. The competition game of individual players is assigned a non-cooperative game which used to presente by extensive and the normal forms.
The extensive form can be used to formalize games involving timed moves On these islands, games are played on trees (Figure 2.3).
Figure 2.3 Extension Form Game The players, strategies, and payoffs of each player in the normal (or strategic form) game is usually represented by a matrix (Figure 2.4).
Figure 2.4 Normal Form Game The formulation of the "prisoner's dilemma" (Flood & Dresher, 1950) and the definition of the Nash equilibrium laid the groundwork for contemporary noncooperative game theory.
The Nash equilibrium solution is applied to solve a noncooperative game containing two or more players, in which the strategies of players are assumed to know by others, however no one may obtain more profits while they change their individual strategies only (Osborne and Rubinstein 1994) In other words, if the player has decided their strategies and no one can receive more gains by altering his own plan if all players do not want to shift their decision, finally the set of strategy choices and the corresponding payoffs obtain the balancing point, it calls the Nash equilibrium The Nash equilibrium contributes a solution for predicting the outcome when all decisions are made at the same time by all participants, and the result is affected by the decision of others In Nash equilibrium problem, the strategy set of each player is exceeding the control of the remaining player's decision. Nevertheless, in some cases, a player's decision may be limited by the others's decision If players own some shared resources or restrictions, this is known as a generalized Nash equilibrium In such situations, all players’s strategies are restricted by common or shared constraints (Facchinei and Kanzow 2010).
A strategy profile is a collection of individual strategies for each player. Informally, a strategy profile is a Nash equilibrium if no player can improve his performance by changing his strategy unilaterally Consider what this means if each player is informed of the strategies of the others Assume then that each player asks himself: "Knowing the other players' strategies and treating them as set in stone, can
I benefit from changing my strategy?"
If any player can respond "Yes," then the set of strategies is not Nash. However, if each player prefers not to switch (or is indifferent between switching and remaining), the strategy profile is Nash Thus, each strategy in a Nash equilibrium is the optimal response to the strategies of the other players.
Formally, let S i be the set of all possible strategies for player i, where i=1,…N Let s*=(s* i , s*- i ) be a strategy profile, a set consisting of one strategy for each player, where s*- i denotes the N - 1 strategies of all the players except i Let u i (s i , s*- i ) be player i's payoff as a function of the strategies The strategy profiles* is a Nash equilibrium if ݑ ݅ ݏ ݅ כ ǡ ݏ െ݅ כ ݑ ݅ ݏ݅ǡ ݏ െ݅ כ ݂ݎ ݈݈ܽ ݏ ݅ א ܵ ݅
A game may contain multiple Nash equilibrium states Even if the equilibrium is unique, it may be unsatisfactory: a player may be indifferent between several strategies given the choices of the other players If the inequality is strict, it is unique and is referred to as a strict Nash equilibrium; thus, one strategy is the unique best response: ݑ ݅ ݏ ݅ כ ǡ ݏ െ݅ כ ݑ ݅ ݏ ݅ ǡ ݏ െ݅ כ ݂ݎ ݈݈ܽ ݏ ݅ א ܵ ݅ ǡ ݏ ݅ ് ݏ ݅ כ Take note that the strategy set Si may vary according to the player, and its elements may consist of a variety of mathematical objects To put it simply, a player may opt for one of two strategies, e.g S i ={Yes, No} Alternatively, the strategy set could be a finite set of conditional strategies that respond to the actions of other players., e.g ܵ ݅ ൌ ܻ݁ݏ ൌ ܮݓǡ ܰ ൌ ܪ݄݅݃ Or, it might be an infinite set, a continuum or unbounded, e.g S i ={Price} such that Price is a non-negative real number While Nash's existence proofs imply the existence of a finite strategy set, the concept of Nash equilibrium does not.
In third-person perspective, the Nash equilibrium may appear irrational at times This is due to the fact that a Nash equilibrium is not always Pareto optimal.
Nash equilibrium may also have irrational consequences in sequential games, as players may "threaten" one another with threats they will never carry out For these games, the subgame perfect Nash equilibrium may be a more useful analytical tool.
2.2.4- Game Theory Applications in Construction Project Management
Game theory has been used to explain and predict outcomes in various facets of construction management The effect of various factors on the relationship between the owner and the contractor, the contractors and the subcontractors.
Among them, Hanaoka [36] integrated theory of Monte Carlo simulation and bargaining game to determine the appropriate concession period, moreover, the effectiveness of proposed model has been demonstrated by case study of two BOT road projects in the Philippines Asgari [37] used Shapley Value solution in cooperative game to maximize the sub-contractor's benefits for each feasible coalition in the construction joint-resource management model Li [38] used the theory of bargaining game in building a model of improved alternating offer, to aim at determining the risk allocation among participants in the construction stage of public-private partnership (PPP) projects Bayat [39] coordinated the theory of fuzzy and theory of bilateral bargaining game to optimize value between the length of concession duration and capital frame in build–operate–transfer (BOT) agreements Shang [40] supported the owner to propose the modeling of payment mechanisms for PPPs related transportation by developing the Stackelberg game model, the model contributes to maximized aim for the overall performance of the project in order to get social benefits while that maximize the profit for the interest of private investment as well.
Ho [41] used game theory to dissect the information asymmetry that existed during the procurement of the BOT project and its implications for project financing and government policy during the preconstruction stage Ahmed [42] used noncooperative game theory to examine — and potentially mitigate — industry exposure to the "winner's curse" in single-stage and multistage construction bidding environments.
According to Abotaleb [43], the proposed model of construction bidding markup estimation is based on a Bayesian analytic framework and assists contractors in increasing their probability of winning projects by optimizing bid prices while maintaining a reasonable profit Xu [44] simulated the model of supplier selection and dynamic inventory problems to aiming at obtaining the satisfactory benefits and attaining equilibrium of both the construction firms and the selected suppliers based on solution combination of Stackelberg game, Nash equilibrium and swarm optimization algorithm Jin [45] understood and provided a framework for the renegotiation process in PPP projects users paid, the aim of developing the bargaining game model is to adjust optimally values between the concession period and government guarantee for the beginning of a renegotiation stage, whereas, concession price model of renegotiation is represented by coalitional game Assaad [46] modeling the bidding-decision making process based on 982 US public construction projects collected and then comparing the performance of three learning algorithmic game theory approaches with two bidding strategies as winning more projects and reducing the winner's curse situation.
Modelling Assumption
Simple assumptions are necessary to perform a game-theoretic study and generate meaningful conclusions Our model set-up and assumptions are as follows:
It is assumed that construction firms entering the bidder are homogeneous in terms of capability and experience Accordingly, the players are representatives of the contractor and owner position in the model of competitive game Likewise, the essential information of project and potential competitors in the market are easy to identify, thus, it can be naturally assumed that the model is complete information. For that reason, the payoff constituted from profit margin and additional effort cost for each player can be reasonably calculated by others.
Marginal profit (P) – The highest profit of the contractor shall be achieved.
The assumed types of effort are high and average levels, denoted by H and A,respectively The effort A is defined as the level of effort that meets the typical requirements from tenderer and does not incur any suggestions for quality improvement purpose Conversely, the effort H is defined as the level of effort that will suggest optimal alternatives to upgrade the quality of a proposal which need to be incurred extra costs, which is shown as E This enhancement is recognized by an effective proposal evaluation system specified in documentation of Request for Proposal (RFP) The evaluation criteria and their respective weights in RFP would be embodied the standard of quality.
The extra cost (E) – For practical purposes, the extra cost (E) is assuming that it would not be embraced in the bid price To win the contract under the price-quality competition, the contractor investing additional effort needs to trade-off between quality improvement and bid price increase This assumption is appropriate in applying the best-value approach In this paper, the amount of E shall be defined as a ratio of P, for instance, E=0.1P.
Bid compensation (S) – Bid compensation is suggested to pay the second rank bidder only Because the owner concerns that a number of participators submit effortless proposals to receive a stipend (DBIA’s manual – 1996b). The amount of S shall be specified as a rational number of E, for example, S=0.5E or S
a – the chance of winning when the contractor employs strategy 'H'
1 - a – the chance of losing when the contractor employs strategy 'H'
b – the chance of winning when the contractor employs strategy 'A' (b ≤ a)
1 - b – the chance of winning when the contractor employs strategy 'A'
It is assumed that the strategy ‘No S’ of the owner may influence to the amount of contractor’ extra effort, denote byჴE (0 0.7 (strong and very strong contractor) and increases as the ratio E/P increases. For a weak contractor (ჴ=0), the owner has a 50% probability of employing strategy 'S', increasing to 100% when the additional effort cost is nearly equal to half of the contractor's profit (E=0.4P). c) The magnitude of project complexity and considering the bidder capability
Model verification: Bid Compensation and Procurement Strategies
The results of this study were verified using the same input scenario as that used by [13] The parameters used in the proposed model were generated based on contractors’ strategical tendencies in past research However, this study was more sophisticated than others because it considered more complicated scenarios and provided owners with an optimized solution for effective bid compensation use.
The amount of additional effort required for amelioration is denoted in this paper by the letter E The amount of additional effort required for amelioration is dependent on the complexity of the project For instance, when E = 2%, as illustrated in Figure 7, bid compensation should be utilized in two cases (4H or 4A); otherwise, when E > (3/5)P, bid compensation is not recommended in most cases.
As a result, we can conclude that offering compensation is inefficient when the project is either extremely simple or extremely complex.
We can conclude from the research findings in Table 3.2 and Figure 1 that the amount of bid compensation has an effect on the likelihood of a contractor playing 'H' According to the numerical results of desired u* based on S variation and taking bidder capability into account, the probability of the bidder exerting a great deal of effort increases as the bid compensation amount increases.
In the case of projects that require high-quality and technical expertise, although bid compensation was regarded more effective, it was not necessarily preferred in practice For example, when E = 5.5% (illustrated in Table 3.2), the probability of high playing efforts were 55% and 66% for S = 6.5% and 7.5%, respectively At the end of the day, marginal cost–benefit analysis should be used to determine whether to use bid compensation, if so, what amount.
Moreover, our analysis of varying strategies in our owner–bidder bid compensation game suggests theoretical principles that might underpin future bid compensation decisions Three notable conclusions regarding the formation of bid compensation strategies are presented as follows: First, although bid compensation has less influence on the behaviour of larger contractors, it markedly affects decision-making among medium- and small-scale contractors The model aims to help the owner assess the relevance and preferred magnitude of bid compensation, denoted as S in this paper.
Table 3.2 - Mixed Strategy Probabilities in a Four-Bidder Game Adapted from
“Bid compensation decision model for projects with costly bid preparation” by Ho [13]
Note: N/A = respective equilibrium does not exist.
H = inputing more efforts; A = no more efforts;
For example: 3H+1A = there are 3 bidders who invest more effort and a bidder is not invest the effort.
Second, apart from project complexity, two other dimensions may affect bid compensation strategies: contractor capacity and the ratio between the probabilities of winning the contract based on whether the contractor invests additional effort. Contractor capacity was classified into three categories, denoted in this paper by the abbreviation ჴ The ratio of the probability of winning after investing additional effort to the probability of winning without investing additional effort is denoted by b/a.
Finally, bid compensation is undesirable when the benefit of selecting 'H' over 'A' is negligible Furthermore, bid compensation is not recommended in cases where only a small amount of additional bidder effort will be generated because the owner outlay involved will not be justified In projects for which E has a large effect on expected profit, bid compensation represents an effective and valuable tool for project stakeholders.
GAME OF BID COMPENSATION FOR
In this chapter, we present the general process to construct game of bid compensation model among the owner and heterogeneous bidder.
It started with introducing the background scenario and necessary simplification assumption affecting bid compensation outcomes
(Section 4.1) The construction and discussion of bid compensation model using noncooperative game is presented in different scenarios which increase the number of players (Section 3.2) The comparison and outcomes of this chapter are explained in detail in Section 4.3.
Background Scenario and Modeling Assumption
In the construction market, varying strengths of construction firms are common as scale, the capability of finance and technical, fame Depending on these characteristics, the companies can be selected by the owner through the pre- qualification process In there, the contractors have experience or knowledge in similar projects that is a remarkable advantage over other competitors Overall, the companies own a huge amount of important resources (material and human), show stronger capability for technology and finance, or get greater renown than competitive others, that is regarded superior, defined as strong contractors in this paper Owners tend to prefer these companies, thus, their chance of bid winning is higher also Another group is normal contractors, thereby, they are less powerful in competition and low the ability of bid winning.
With the current great stride in information technology, the information of projects and bidders are easily acquired Furthermore, for large-scale projects, the number of contractors in bidding is small because of high initial requirements, hence, the owner and bidders know clearly the attendee about the power of competition, imperfection, the expected profit, and project cost Consequently, it is reasonable for assuming complete information modeling Under completeness of information, each player identifies other competitors who are strong or regular, as well as, they also calculate their project payoffs and others.
4.1.1- Bidding strategies of Bidders and Owner
For bidders, the assumed effort levels are high and average, denoted by the letters H and A, respectively Level A effort is defined as the level of effort that satisfies the tenderee's typical requirements without requiring any suggestions for quality improvement In contrast, the level H of effort is defined as the level at which an alternative proposal is prepared and an additional cost, denoted by the letter E, is incurred in order to improve the quality of a proposal, as determined by an effective proposal evaluation system Technically, the requirement's evaluation criteria and their associated weights have been converted from the standard of quality.
For the owner, it has two strategies including compensation and no bid compensation, denoted by S and No S, respectively The S strategy is compensation offered by the owner to a second-rank bidder for their extra effort in the proposal.
On the contrary, the No S strategy is not applied compensation for the unsuccessful bidder.
The model's payoffs are determined by the following components: (1) P – The highest profit of the contractor shall be achieved; (2) S – Bid compensation is assumed for the second rank bidder; (3)E– For simplicity, it is assumed that E will be decoupled from the tender price, implying that the contractor must consider price-quality competition in order to win the contract for performing high effort; (4) ჴ – it is assumed that if the Owner chooses strategy “no S” so bidder play extra effort with ratioჴ(ჴE).
Model setup and Discussion
4.2.1- Modelling of 3 players: 1 owner – 1 strong bidder – 1 regular bidder
It is considered that there are 8 possible Nash equilibrium for an owner, strong and regular bidders as (S, H, H), (S, H, A), (S, A, H), (S, A, A), (No S, H, H), (No S, H, A), (No S, A, H) and (No S, A, A) Due to the strong bidder takes advantage to compare with the regular bidder and always wins the bid (by 100% of probability) if they choose the same level of effort (H or A) or strong bidder play with level H Regular bidder has a winning chance with a 50% of probability at (A, H) case Thus, the possible payoffs of each player are shown in table 4.1.
Table 4.1 The expected payoffs for Three-Players with Owner, A Strong Bidder, A
4.2.1.1- Pure Strategy Nash Equilibrium (PSNE)
The Owner does not wander from [(S; H; H), (E – S; P – E; S – E)] to [(No S; H; H), (ჴE; P – E; -ჴE)]
The strong bidder does not wander from [(S; H; H), (E – S; P – E; S – E)] to [(S; A; H), (E/2 – S; P/2 + S/2; P/2 + S/2 – E)]
The regular bidder does not wander from [(S; H; H), (E – S; P – E; S – E)] to [(S; H; A), (E – S; P – E; S)]
To execute the aforementioned logic for other equilibriums (please see the detail in appendix), it is found that no pure equilibrium can be exist according to the results shown in table 4.2
Table 4.2 The summary of checking pure strategy Nash Equilibrium for Model of Three-Players with Owner, One Strong Bidder, One Regular Bidder
4.2.1.2- Mixed Strategy Nash Equilibrium (MSNE)
Considering all players playing with mixed strategy, v*, u*, and w* are probability equilibrium of owner, strong bidder, regular bidder, respectively The MSNE requirement is obtained in three equations as follows.
Balancing Owner’s payoffs Ȃ כ כ ȀʹȂ ͳȂ כ כ Ȃ כ ͳȂ כ െ ͳȂ כ ͳȂ כ ൌ ჴ כ כ ჴȀʹሺͳ Ȃ כ ሻ כ ჴ כ ሺͳ Ȃ כ ሻ Ͳሺͳ Ȃ כ ሻሺͳ Ȃ כ ሻ ֜ ሾሺͳ Ȃ ჴሻ Ȃ ሿ כ כ ሾȀʹሺͳ Ȃ ჴሻ Ȃ ሿሺͳ Ȃ כ ሻ כ ሾሺͳ െ ჴሻ Ȃ ሿ כ ሺͳ Ȃ כ ሻ ሺ െ ሻሺͳ Ȃ כ ሻሺͳ Ȃ כ ሻ ൌ Ͳ ֜ כ ʹ כ െ כ כ ൌ ʹȀሾሺͳ െ ჴሻሿ (Eq 4.4) Balancing Strong bidder’s payoffs Ȃ כ כ Ȃჴ ͳȂ כ כ Ȃ כ ͳȂ כ Ȃჴ ͳȂ כ ͳȂ כ ൌ ሺȀʹ Ȁʹሻ כ כ Ȁʹሺͳ Ȃ כ ሻ כ כ ሺͳ Ȃ כ ሻ ሺͳ Ȃ כ ሻሺͳ Ȃ כ ሻ ֜ ሺȀʹ Ȃ Ȁʹ Ȃ ሻ כ כ ሺȀʹ Ȃ ჴሻሺͳ Ȃ כ ሻ כ ሺ െ ሻ כ ሺͳ Ȃ כ ሻ ሺ െ ჴሻሺͳ Ȃ כ ሻሺͳ Ȃ כ ሻ ൌ Ͳ ֜ כ Ȁ Ȃ כ כ Ȁ Ȃ ʹሺͳ Ȃ ჴሻ כ ൌ ʹჴ (Eq 4.5) Balancing Regular bidder’s payoffs Ȃ כ כ Ȁʹ ȀʹȂ כ ͳȂ כ െჴ ͳȂ כ כ Ȁ ʹȂჴ ͳȂ כ ͳȂ כ ൌ כ כ כ ሺͳ Ȃ כ ሻ Ͳሺͳ Ȃ כ ሻ כ Ͳሺͳ Ȃ כ ሻሺͳ Ȃ כ ሻ ֜ ሺ െ ሻ כ כ ሺȀʹȂȀʹȂሻ כ ሺͳȂ כ ሻ ሺ െ ჴሻሺͳȂ כ ሻ כ ሺȀ ʹȂჴሻሺͳȂ כ ሻሺͳȂ כ ሻ ൌ Ͳ ֜ כ כ Ȁ Ȃ ሾȀ ʹሺͳ Ȃ ჴሻሿ כ Ȃ כ Ȁ ሺȀ Ȃ ʹჴሻ ൌ Ͳ (Eq 4.6)
According to Figure 4.1, the strong bidder always tends to play full effort by 100% probability whether bid compensation and its amount or not The reason is that the strong bidder would like to hold the winning chance steady in a game of bidding In contrast, the amount of bid compensation impacts to regular bidders in inputting extra effort decisions, and probability can increase by 15% from S=E to S (Figure 4.1).
Figure 4.1 The chance of playing high effort of bidders
On the other hand, when an owner sees that the strong bidder always chooses strategy H as result in Figure 4.1, the Eq 4.7 is solved as below. ʹȀ ͳ െ Ƚ ൌ כ ʹ כ െ כ כ (Eq 4.7) When u* = 1, the maximum value of (w* + 2u* - u*w*) is 2 without considering the amount of w*, it means strong bidder playing full effort, the regular bidder can use high or average effort strategy.
ჴ = 1, meaning that all bidders play full effort whether a strategy of bid compensation by the owner or not, thus, the owner should not offer compensation.
ჴ= 0, meaning that the bidder won’t input extra effort if bid compensation is not applied For that reason, the owner should consider using bid compensation and its amount by S = E to encourage the regular bidder.
ჴ = 0.5 => S/E = 1/2, meaning that if bidder inputs a half extra effort as owner’s desire, the owner considers bid compensation with 50% of probability and its amount by S = E.
4.2.2- Modelling of Four-players: 1 owner – 2 strong bidders – 1 regular bidder
The modeling of four-players are 16 possible Nash Equilibrium included (S,2H, H), (S, H, A, H), (S, A, H, H), (S, 2A, H), (S, 2H, A), (S, H, A, A), (S, A, H, A),(S, 2A, A), (No S, 2H, H), (No S, H, A, H), (No S, A, H, H), (No S, 2A, H), (No S,2H, A), (No S, H, A, A), (No S, A, H, A), (No S, 2A, A) The expected payoffs are shown in table 4.3.
Table 4.3 The expected payoffs for Four-Players with Owner, Two Strong Bidders,
4.2.2.1- Pure Strategy Nash Equilibrium (PSNE)
The equilibrium conditions' procedure is solved similarly with the three- players game shown previously Since the length of a paper is limited, all calculations for an equilibrium of each player are presented in appendix 2 The result of pure strategy Nash Equilibrium can be existed as table 4.4.
Table 4.4 The summary of checking pure strategy Nash Equilibrium for Model of Four-Players with Owner, Two Strong Bidders, One Regular Bidder
4.2.2.2- Mixed Strategy Nash Equilibrium (MSPE)
Logical similarity with the above three-players model, v*, u*,w* and t* are probability equilibrium of owner, two strong bidders, a regular bidder, respectively. The MSNE requirement is obtained in four equations as following.
Balancing Owner’s payoffs െ כ כ כ ͳȂ כ כ כ כ ͳȂ כ כ כ כ ͳȂ כ ͳȂ כ כ ͳȂ כ כ ͳȂ כ ͳȂ כ Ȁ͵ െ ͳȂ כ ͳȂ כ כ െ ͳȂ כ ͳȂ כ ͳȂ כ ൌ ჴ כ כ כ ͳȂ כ כ כ כ ͳȂ כ כ כ כ ͳȂ כ ͳȂ כ כ ͳȂ כ כ ͳȂ כ ͳȂ כ ჴȀ͵ ͳȂ כ ͳȂ כ כ Ͳ ͳȂ כ ͳȂ כ ͳȂ כ ֜ ሾሺͳȂჴሻȂȀሿ כ כ כ ͳȂ כ כ כ כ ͳȂ כ כ כ כ ͳȂ כ ͳȂ כ כ ͳȂ כ כ ͳȂ כ ͳȂ כ ሾͳȀ͵ሺͳȂჴሻȂȀሿ ͳȂ כ ͳȂ כ כ ሺ െ Ȁ ሻ ͳȂ כ ͳȂ כ ͳȂ כ ൌ Ͳ (Eq 4.8)
Balancing 1 st Strong Bidder’s payoffs Ȁʹ ȀʹȂ כ כ כ כ כ ͳȂ כ Ȃ כ ͳȂ כ כ כ ͳȂ כ ͳȂ כ ȀʹȂჴ ͳȂ כ כ כ ͳȂ כ כ ͳȂ כ Ȃჴ ͳȂ כ ͳȂ כ כ ͳȂ כ ͳȂ כ ͳȂ כ ൌ Ȁʹ כ כ כ Ȁ͵ Ȁ͵ כ ͳȂ כ כ כ כ ͳȂ כ Ȁʹ Ȁʹ כ ͳȂ כ ͳȂ כ Ͳ ͳȂ כ כ כ ͳȂ כ כ ͳȂ כ Ȁ ͵ ͳȂ כ ͳȂ כ כ Ȁʹ ͳȂ כ ͳȂ כ ͳȂ כ ֜ ȀʹȂͳ כ כ כ ȀʹȂȀʹȂͳ כ כ ͳȂ כ ʹȀ͵ȂȀ ͵Ȃͳ כ ͳȂ כ כ ȀʹȂȀʹȂͳ כ ͳȂ כ ͳȂ כ ȀʹȂჴ ͳȂ כ כ כ כ ͳȂ כ ͳȂ כ ͳȂ כ ʹȀ͵Ȃჴ ͳȂ כ ͳȂ כ כ ൌ Ͳ ֜ ȀʹȂͳ כ כ כ ȀʹȂȀʹȂͳ כ כ ͳȂ כ ʹȀ͵ȂȀ ͵Ȃͳ כ ͳȂ כ כ ȀʹȂȀʹȂͳ כ ͳȂ כ ͳȂ כ ȀʹȂჴ ͳȂ כ כ כ כ ͳȂ כ ͳȂ כ ͳȂ כ ʹȀ͵Ȃჴ ͳȂ כ ͳȂ כ כ ൌ Ͳ (Eq 4.9) Balancing 2 nd Strong Bidder’s payoffs Ȁʹ ȀʹȂ כ כ כ כ כ ͳȂ כ Ȃ כ ͳȂ כ כ כ ͳȂ כ ͳȂ כ ȀʹȂჴ ͳȂ כ כ כ ͳȂ כ כ ͳȂ כ Ȃჴ ͳȂ כ ͳȂ כ כ ͳȂ כ ͳȂ כ ͳȂ כ ൌ Ȁʹ כ כ כ Ȁ͵ Ȁ͵ כ ͳȂ כ כ כ כ ͳȂ כ Ȁ ʹ Ȁʹ כ ͳȂ כ ͳȂ כ Ͳ ͳȂ כ כ כ ͳȂ כ כ ͳȂ כ Ȁ ͵ ͳȂ כ ͳȂ כ כ Ȁʹ ͳȂ כ ͳȂ כ ͳȂ כ ֜ ȀʹȂͳ כ כ כ ȀʹȂȀʹȂͳ כ כ ͳȂ כ ʹȀ͵ȂȀ͵Ȃͳ כ ͳȂ כ כ ȀʹȂȀʹȂͳ כ ͳȂ כ ͳȂ כ ȀʹȂჴ ͳȂ כ כ כ Ȁ ʹȂჴ ͳȂ כ כ ͳȂ כ ʹȀ͵Ȃჴ ͳȂ כ ͳȂ כ כ Ȁ ʹȂჴ ͳȂ כ ͳȂ כ ͳȂ כ ൌ Ͳ (Eq 4.10)
Balancing Regular Bidder’s payoffs ሺ െ ሻ כ כ כ ሺȀʹȂሻሾ כ ሺͳȂ כ ሻ כ כ כ ሺͳȂ כ ሻሿ ሺȀ͵ Ȁ ͵ Ȃ ሻ כ ሺͳȂ כ ሻሺͳȂ כ ሻ ሺ െ ჴሻሺͳ Ȃ כ ሻሾሺ כ כ ሺͳȂ כ ሻ כ כ ሺͳȂ כ ሻሿ ሺȀ ͵ Ȃ ჴሻሺͳ Ȃ כ ሻሺͳȂ כ ሻሺͳȂ כ ሻ ൌ Ͳ ֜ ሺ െ ͳሻ כ כ כ ሺȀʹ Ȃ ͳሻሾ כ ሺͳȂ כ ሻ כ כ כ ሺͳȂ כ ሻሿ ሺȀ͵ Ȁ ͵ Ȃ ͳሻ כ ሺͳȂ כ ሻሺͳȂ כ ሻ ሺ െ ჴሻሺͳ Ȃ כ ሻሾሺ כ כ ሺͳȂ כ ሻ כ כ ሺͳȂ כ ሻሿ ሺȀ ͵ Ȃ ჴሻሺͳ Ȃ כ ሻሺͳȂ כ ሻሺͳȂ כ ሻ ൌ Ͳ (Eq 4.11)
Figure 4.2 The chance of playing high effort of biddersFollowing to Figure 4.2, in the model with two strong bidders, the probability of regular bidder is always zero for S=E to S It indicates that the regular bidder is unwilling to input more effort in the proposal, even, when the owner offers the amount double extra cost (S.) but it still can not impact to regular bidder's decision due to the lower competition level than strong bidders For strong bidders in Figure 4.2, the probability of playing high effort is over 90% forS=E to S at the case of E/P=0.05, and it tends to drastically decrease to 12% and 50% for S=E and S=1.5E, respectively However, with S., the 100% probability is unchanged from E/P = 0.05 to 0.45 From that point, it is found that the bid compensation and its amount are significant impacts to strong bidder’s strategy in the game having two strong bidders Based on the desired probability in figure 4.2, the owner can define how much is affordable bid compensation to correspond with the level of project complexity.
Figure 4.3 The probability of bid compensation strategy of the owner,
Figure 4.4 The probability of bid compensation strategy of the owner, E/P=0.45
According to Figure 4.3, with the case E/P=0.35, the probability of bid compensation strategy (v*) ofჴ = 0.1 are 95%, 80% and 70% for S=E, S=1.5E andS., respectively Similarity, with the case E/P=0.45, which is shown in Figure4.4, the v* of ჴ = 0.1 are 73%, 62% and 53% for S=E, S=1.5E and S.,respectively Besides, the v* tends to decrease when ჴ increases to 0.9 and sharply fall at E/P=0.45 comparing to E/P=0.35, which means the probability of bid compensation strategy has drop trend if the difference of putting more effort in bidder’s proposal between S and No S strategy is trivial The chance of using bid compensation by the owner tends to decrease in the condition of the increasing amount of S and the increasing of E/P ratio It is concluded that the bid compensation strategy is influenced by the complex level of the project and ratio factor with the extra cost (E).
4.2.3- Modelling of four-players: 1 owner – 1 strong bidder – 2 regular bidders
The modeling of four-players are 16 possible Nash Equilibrium included (S, 2H, H), (S, H, A, H), (S, A, H, H), (S, 2A, H), (S, 2H, A), (S, H, A, A), (S, A, H, A), (S, 2A, A), (No S, 2H, H), (No S, H, A, H), (No S, A, H, H), (No S, 2A, H), (No S, 2H, A), (No S, H, A, A), (No S, A, H, A), (No S, 2A, A) The expected payoffs are shown in table 4.5.
Table 4.5.The expected payoffs for Four-Players with Owner, Two Strong Bidders, One Regular Bidder
4.2.3.1- Pure Strategy Nash Equilibrium (PSNE)
The equilibrium conditions' procedure is solved similarly with the three- players game shown previously Since the length of a paper is limited, all calculations for an equilibrium of each player are presented in appendix 3 The result of pure strategy Nash Equilibrium can be existed as table 4.6.
Table 4.6 The summary of checking pure strategy Nash Equilibrium for Model of
Four-Players with Owner, One Strong Bidder, Two Regular Bidders
4.2.3.2- Mixed Strategy Nash Equilibrium (MSPE)
Logical similarity with the above three-players model, v*, u*, w* and t* are probability equilibrium of owner, a strong bidders, two regular bidders, respectively. The MSNE requirement is obtained in four equations as following.
Balancing Owner’s payoffs Ȃ כ כ כ כ ͳȂ כ כ כ כ ͳȂ כ כ ͳȂ כ ͳȂ כ ʹȀ ͵Ȃ ͳȂ כ כ כ ȀʹȂ ͳȂ כ ͳȂ כ כ ͳȂ כ כ ͳȂ כ െ ͳȂ כ ͳȂ כ ͳȂ כ ൌ Ƚሾ כ כ כ כ ͳȂ כ כ כ כ ͳȂ כ כ ͳȂ כ ͳȂ כ ሿ ʹȽȀ ͵ ͳȂ כ כ כ ȽȀʹ ͳȂ כ ͳȂ כ כ ͳȂ כ כ ͳȂ כ ሿ ሺ െ Ͳሻ ͳȂ כ ͳȂ כ ͳȂ כ ֜ ሾሺͳ Ȃ Ƚሻ Ȃ Ȁሿ כ ሾʹȀ͵ሺͳ Ȃ Ƚሻ Ȃ Ȁሿሺͳ Ȃ כ ሻ כ כ ሾͳȀʹሺͳ Ȃ Ƚሻ Ȃ Ȁ ሿሺͳ Ȃ כ ሻሾሺͳ Ȃ כ ሻ כ כ ሺͳ Ȃ כ ሻሿ ൌ Ͳ (Eq 4.12) Balancing Strong Bidder’s payoffs ሺ Ȃ ሻ כ ሾ כ כ ሺͳ Ȃ כ ሻ כ כ ሺͳ Ȃ כ ሻ ሺͳ Ȃ כ ሻሺͳ Ȃ כ ሻሿ ሺ Ȃ Ƚሻሺͳ Ȃ כ ሻሾ כ כ ሺͳ Ȃ כ ሻ כ כ ሺͳ Ȃ כ ሻ ሺͳ Ȃ כ ሻሺͳ Ȃ כ ሻሿ ൌ ሺȀ͵ Ȁ͵ሻ כ כ כ ሺȀʹ Ȁʹሻ כ ሾሺͳ Ȃ כ ሻ כ כ ሺͳ Ȃ כ ሻሿ כ ሺͳ Ȃ כ ሻሺͳ Ȃ כ ሻ Ȁ͵ሺͳ Ȃ כ ሻ כ כ Ȁʹሺͳ Ȃ כ ሻሾሺͳ Ȃ כ ሻ כ כ ሺͳ Ȃ כ ሻሿ ሺͳ Ȃ כ ሻሺͳ Ȃ כ ሻሺͳ Ȃ כ ሻ ֜ ሼሺʹȀ͵ȂȽሻȂ כ ሾሺͳȂȽሻ Ȁ͵ሿሽ כ כ ሼሺȀʹȂȽሻȂ כ ሾȀʹ ሺͳȂȽሻሿሽሾሺͳȂ כ ሻ כ כ ሺͳȂ כ ሻሿȂሺͳȂ כ ሻሺͳȂ כ ሻሾ כ ȽሺͳȂ כ ሻሿ ൌ Ͳ (Eq 4.13) Balancing 1 st Regular Bidder’s payoffs ሺȀʹȂሻ כ כ כ ሺȀ͵ Ȁ͵Ȃሻ כ ሺͳȂ כ ሻ כ ሺȀʹȂሻ כ כ ሺͳȂ כ ሻ ሺȀʹ Ȁ ʹȂሻ כ ሺͳȂ כ ሻሺͳȂ כ ሻȂჴሺͳ Ȃ כ ሻ כ כ ሺȀ ͵Ȃჴሻሺͳ Ȃ כ ሻሺͳ Ȃ כ ሻ כ Ȃჴሺͳ Ȃ כ ሻ כ ሺͳ Ȃ כ ሻ ሺȀʹȂჴሻሺͳ Ȃ כ ሻሺͳ Ȃ כ ሻሺͳ Ȃ כ ሻ ൌ Ȁʹ כ ሾ כ ሺͳȂ כ ሻ ሺͳȂ כ ሻሺͳȂ כ ሻሿ ֜ ሺȀʹ Ȃ ͳሻ כ כ ሾሺȀ͵ Ȃ Ȁሻ כ Ȁʹ Ȃ ͳሿ כ ሺͳ Ȃ כ ሻ Ȃ ჴሺͳ Ȃ כ ሻ כ ሺͳ Ȃ כ ሻሺͳ Ȃ כ ሻሾሺȀʹ Ȃ ჴሻ Ȃ כ Ȁሿ ൌ Ͳ (Eq 4.14) Balancing 2 nd Regular Bidder’s payoffs ሺȀʹȂሻ כ כ כ ሺȀ͵ Ȁ͵ Ȃሻ כ ሺͳȂ כ ሻ כ ሺȀʹȂሻ כ כ ሺͳȂ כ ሻ ሺȀʹ Ȁ ʹȂሻ כ ሺͳȂ כ ሻሺͳȂ כ ሻȂჴሺͳȂ כ ሻ כ כ ሺȀ ͵ȂჴሻሺͳȂ כ ሻ כ כ ȂჴሺͳȂ כ ሻ כ ሺͳȂ כ ሻ ሺȀʹȂჴሻሺͳȂ כ ሻሺͳȂ כ ሻሺͳȂ כ ሻ ൌ Ȁ ʹ כ ሾ כ ሺͳȂ כ ሻ ሺͳȂ כ ሻሺͳȂ כ ሻሿ ֜ ሺȀʹȂͳሻ כ כ ሾሺȀ͵ ȂȀሻ כ Ȁʹ Ȃͳሿ כ ሺͳȂ כ ሻȂჴሺͳȂ כ ሻ כ ሺͳȂ כ ሻሺͳȂ כ ሻሾሺȀʹȂჴሻȂ כ Ȁሿ ൌ Ͳ (Eq 4.15)
As Figure 4.5, the strong bidder has tendency of playing high effort without considering the bid compensation, and the probability decreases according to the increase of ratio E/P (mean the project becomes complex) In contrary, the regular bidder’s decision is impacted by the increasing of project complexity or the amount of bid compensation significantly.
Figure 4.5 Comparison regular bidder behavior in three-players game and four-players game
Conclusion
The purpose of this paper was to assess the effectiveness of bid compensation strategies from both the owner and contractor perspectives For the owner, misjudgement of the effectiveness of bid compensation results in the loss of an opportunity to implement a measure that encourages greater effort in project planning and tender preparation Additionally, if contractors misunderstand the import of bid compensation, they forfeit the opportunity to receive a stipend to offset the costs associated with tender preparation.
On the basis of game-theoretic analysis, this study developed a model of bid compensation The effectiveness of bid compensation evaluation is contingent upon a variety of factors in various project procurement scenarios This model is based on a unique set of presumptions discussed in each chapter 3 or 4—such as homogeneous or heterogeneous bidders and complete information—and is used to analyze economic behavior among participants and draw useful conclusions from complex scenarios As a result, the model should be considered appropriate for use in a limited number of circumstances Additionally, the model is designed to benefit both the owner and contractors The findings indicated that there are patterns in the relationship between the P, S, and E parameters.
The likelihood of a bidder making high efforts is determined by their capability, the bid compensation amount in relation to the cost of additional effort,and the advantage differential between strategies 'H' and 'A' As a result, owners are advised to employ bid compensation strategies exclusively for weaker bidders and to define the quantity of bid compensation in order to obtain the best effect The intricacy of the project and the capacity of the contractor are two critical criteria that determine an owner's chances of adopting a bid compensation plan Our analysis reveals that if bid compensation is established, the contractor can confidently invest significant effort in more difficult projects.
Future Research Direction
This model is under development and needs some modifications to enhance its features Thus, applying this model opens up many areas of research.
1) We recommend that future research be conducted with a broader scope by relaxing the strict assumptions made in our study Additionally, the current study's empirical survey should be updated to assess the efficacy of incentive mechanisms in promoting higher proposal quality.
2) Due to the time limitation, we only considered the limited number of bidders in bidding process There are still plenty of general model with m-strong bidders and n-regular bidders to figure out the whole system effective performance of proposed model.
3) To our knowledge, the formulation and study of a bidding process in the absence of opponent information have not been thoroughly investigated. Thus, several future research directions can be considered in order to enhance the proposed model's comprehensiveness and realism.
4) Finally, future research could expand on this area by considering alternative approaches, such as calculating payoffs using artificial intelligence This will allow for a more comprehensive examination of the effectiveness of bid compensation while also ensuring the results' generalizability.
[1] R Awwad and M Ammoury, “Owner’s Perspective on Evolution of Bid Prices under Various Price-Driven Bid Selection Methods,” J Comput Civ. Eng., vol 33, no 2, p 04018061, 2019, doi: 10.1061/(asce)cp.1943-
[2] D Kashiwagi and R E Byfield, “Selecting the best contractor to get performance: On time, on budget, meeting quality expectations,” J Facil. Manag., vol 1, no 2, pp 103–116, 2002, doi: 10.1108/14725960310807872.
[3] W Art Chaovalitwongse, W Wang, T P Williams, and P Chaovalitwongse,
“Data Mining Framework to Optimize the Bid Selection Policy for Competitively Bid Highway Construction Projects,” J Constr Eng Manag., vol 138, no 2, pp 277–286, 2012, doi: 10.1061/(asce)co.1943-7862.0000386. [4] W Lo, C L Lin, and M R Yan, “Contractor’s Opportunistic Bidding Behavior and Equilibrium Price Level in the Construction Market,”J Constr. Eng Manag., vol 133, no 6, pp 409–416, 2007, doi: 10.1061/(asce)0733-
[5] B G Hwang, S R Thomas, C T Haas, and C H Caldas, “Measuring the Impact of Rework on Construction.pdf,”J Constr Eng Manag., vol 135, no.
March, pp 187–198, 2009, [Online] Available: https://ascelibrary.org/doi/10.1061/%28ASCE%290733-
[6] J L Burati, J J Farrington, and W B Ledbetter, “Causes of Quality Deviations in Design and Construction,”J Constr Eng Manag., vol 118, no.
[7] P E Eriksson and M Westerberg, “Effects of cooperative procurement procedures on construction project performance: A conceptual framework,”
Int J Proj Manag., vol 29, no 2, pp 197–208, 2011, doi:
[8] KPMG, “PPP Procurement Handbook,” no 2010 KPMG, an Australian partnership and a member firm of the KPMG network of independent member firms affiliated with KPMG International Cooperative (“KPMG International”), a Swiss entity, 2010.
[9] P Carrillo, H Robinson, P Foale, C Anumba, and D Bouchlaghem,
“Participation, Barriers, and Opportunities in PFI: The United Kingdom Experience,” J Manag Eng., vol 24, no 3, pp 138–145, 2008, doi:
[10] T Dixon, G Pottinger, and A Jordan, “Lessons from the private finance initiative in the UK: Benefits, problems and critical success factors,”J Prop. Invest Financ., vol 23, no 5, pp 412–423, 2005, doi:
[11] A P C Chan, P T I Lam, D W M Chan, E Cheung, and Y Ke,
“Potential Obstacles to Successful Implementation of Public-Private Partnerships in Beijing and the Hong Kong Special Administrative Region,”J. Manag Eng., vol 26, no 1, pp 30–40, 2010, doi: 10.1061/(asce)0742-
[12] M A Soomro and X Zhang, “Evaluation of the Functions of Public Sector Partners in Transportation Public-Private Partnerships Failures,” J Manag. Eng., vol 32, no 1, p 04015027, 2016, doi: 10.1061/(asce)me.1943-
[13] S P Ho, “Bid compensation decision model for projects with costly bid preparation,”J Constr Eng Manag., vol 131, no 2, pp 151–159, 2005, doi:
[14] S P Ho and Y Hsu, “Bid compensation theory and strategies for projects with heterogeneous bidders: A game theoretic analysis,”J Manag Eng., vol.
[15] D De Clerck and E Demeulemeester, “Creating a More Competitive PPP Procurement Market: Game Theoretical Analysis,” J Manag Eng., vol 32, no 6, p 04016015, 2016, doi: 10.1061/(asce)me.1943-5479.0000440.
[16] K Al-Reshaid and N Kartam, “Design-build pre-qualification and tendering approach for public projects,” Int J Proj Manag., vol 23, no 4, pp 309–
[17] Z Hatush and M Skitmore, “Contractor Selection Using Multicriteria Utility Theory: An Additive Model,” Build Environ., vol 33, no 2–3, pp 105–115,
[18] D J Watt, B Kayis, and K Willey, “The relative importance of tender evaluation and contractor selection criteria,” Int J Proj Manag., vol 28, no.
[19] S Tai, Y Wang, and C J Anumba, “A survey on communications in large- scale construction projects in China,” Eng Constr Archit Manag., vol 16, no 2, pp 136–149, 2009, doi: 10.1108/09699980910938019.
[20] A K Birjandi, F Akhyani, R Sheikh, and S S Sana, “Evaluation and selecting the contractor in bidding with incomplete information using MCGDM method,” Soft Comput., vol 23, no 20, pp 10569–10585, 2019, doi: 10.1007/s00500-019-04050-y.
[21] G Arslan, S Kivrak, M T Birgonul, and I Dikmen, “Improving sub- contractor selection process in construction projects: Web-based sub- contractor evaluation system (WEBSES),”Autom Constr., vol 17, no 4, pp.
[22] S C Ngai, D S Drew, H P Lo, and M Skitmore, “A theoretical framework for determining the minimum number of bidders in construction bidding competitions,” Constr Manag Econ., vol 20, no 6, pp 473–482, 2002, doi:
[23] M.-L Wu and H.-P Lo, “Optimal Strategy Modeling for Price-Time Biparameter Construction Bidding,” J Constr Eng Manag., vol 135, no 4, pp 298–306, 2009, doi: 10.1061/(asce)0733-9364(2009)135:4(298).
[24] A Leśniak and E Plebankiewicz, “Modeling the Decision-Making Process Concerning Participation in Construction Bidding,” J Manag Eng., vol 31, no 2, p 04014032, 2015, doi: 10.1061/(asce)me.1943-5479.0000237.
[25] H M Al-Humaidi, “Construction projects bid or not bid approach using the fuzzy technique for order preference by similarity FTOPSIS method,” J. Constr Eng Manag., vol 142, no 12, 2016, doi: 10.1061/(ASCE)CO.1943-
[26] G Li, C Chen, I Martek, Y Ao, and J Dai, “Bid or No-Bid Decision Model for International Construction Projects: Evidential Reasoning Approach,” J. Constr Eng Manag., vol 147, no 2, p 04020161, 2021, doi:
[27] R A Ojelabi, O O Oyeyipo, A O Afolabi, and I O Omuh, “Evaluating barriers inhibiting investors participation in Public-Private Partnership project bidding process using structural equation model,” Int J Constr Manag., vol.
[28] M Li, M Baek, and B Ashuri, “Forecasting Ratio of Low Bid to Owner’s Estimate for Highway Construction,” J Constr Eng Manag., vol 147, no 1, p 04020157, 2021, doi: 10.1061/(asce)co.1943-7862.0001970.
[29] R Sugden, “Hume’s experimental psychology and the idea of erroneous preferences,” J Econ Behav Organ., vol 183, pp 836–848, 2021, doi:
[30] S M Je and J H Huh, “Estimation of future power consumption level in smart grid: Application of fuzzy logic and genetic algorithm on big data platform,” Int J Commun Syst., vol 34, no 2, pp 1–22, 2021, doi:
[31] Y Wang, Z Bu, H Yang, H J Li, and J Cao, “An effective and scalable overlapping community detection approach: Integrating social identity model and game theory,” Appl Math Comput., vol 390, p 125601, 2021, doi:
[32] M K Sohrabi and H Azgomi, “A Survey on the Combined Use of Optimization Methods and Game Theory,”Arch Comput Methods Eng., vol.
[33] X H Xing, Z H Hu, and W P Luo, “Using evolutionary game theory to study governments and logistics companies’ strategies for avoiding broken cold chains,”Ann Oper Res., 2020, doi: 10.1007/s10479-020-03599-4.
[34] W Amin, Q Huang, M Afzal, A A Khan, K Umer, and S A Ahmed, “A converging non-cooperative & cooperative game theory approach for stabilizing peer-to-peer electricity trading,”Electr Power Syst Res., vol 183, no 2017, p 106278, 2020, doi: 10.1016/j.epsr.2020.106278.
[35] C Esposito, O Tamburis, X Su, and C Choi, “Robust Decentralised Trust Management for the Internet of Things by Using Game Theory,”Inf Process. Manag., vol 57, no 6, p 102308, 2020, doi: 10.1016/j.ipm.2020.102308.
[36] S Hanaoka and H P Palapus, “Reasonable concession period for build- operate-transfer road projects in the Philippines,”Int J Proj Manag., vol 30, no 8, pp 938–949, 2012, doi: 10.1016/j.ijproman.2012.02.001.
[37] S Asgari, A Afshar, and K Madani, “Cooperative Game Theoretic Framework for Joint Resource Management in Construction,”J Constr Eng. Manag., vol 140, no 3, p 04013066, 2014, doi: 10.1061/(asce)co.1943-
[38] Y Li, X Wang, and Y Wang, “Using Bargaining Game Theory for Risk Allocation of Public-Private Partnership Projects: Insights from Different Alternating Offer Sequences of Participants,” J Constr Eng Manag., vol.
143, no 3, p 04016102, 2017, doi: 10.1061/(asce)co.1943-7862.0001249. [39] M Bayat, M Khanzadi, F Nasirzadeh, and A Chavoshian, “Financial conflict resolution model in BOT contracts using bargaining game theory,”
Constr Innov., vol 20, no 1, pp 18–42, 2020, doi: 10.1108/CI-12-2018-
[40] L Shang and A M Abdel Aziz, “Stackelberg Game Theory-Based Optimization Model for Design of Payment Mechanism in Performance- Based PPPs,” J Constr Eng Manag., vol 146, no 4, pp 1–13, 2020, doi:
[41] S P Ho, “Real Options and Game Theoretic Valuation, Financing andTendering For Investments on Built-Operate-Transfer Projects.pdf,” Ph.D.Dissertation, Univ of Illinois at Urbana–Champaign, Urbana, Ill., 2001.
[42] M O Ahmed, I H El-Adaway, K T Coatney, and M S Eid, “Construction bidding and the winner’s curse: Game theory approach,” J Constr Eng. Manag., vol 142, no 2, pp 1–9, 2016, doi: 10.1061/(ASCE)CO.1943-
[43] I S Abotaleb and I H El-Adaway, “Construction Bidding Markup Estimation Using a Multistage Decision Theory Approach,” J Constr Eng. Manag., vol 143, no 1, pp 1–18, 2017, doi: 10.1061/(ASCE)CO.1943-
[44] J Xu and S Zhao, “Noncooperative Game-Based Equilibrium Strategy to Address the Conflict between a Construction Company and Selected Suppliers,” J Constr Eng Manag., vol 143, no 8, p 04017051, 2017, doi:
[45] H Jin, S Liu, J Li, and C Liu, “A game-theoretic approach to developing a concession renegotiation framework for user-pays PPPs,” Int J Constr. Manag., vol 20, no 6, pp 642–652, 2020, doi: 10.1080/15623599.2020.1738003.
[46] R Assaad, M O Ahmed, I H El-adaway, A Elsayegh, and V S Siddhardh Nadendla, “Comparing the Impact of Learning in Bidding Decision-Making Processes Using Algorithmic Game Theory,”J Manag Eng., vol 37, no 1, p.
[47] S P Ho, “Government Policy on PPP Financial Issues: Bid Compensation and Financial Renegotiation,” Policy, Financ Manag Public-PrivatePartnerships, pp 267–300, 2009, doi: 10.1002/9781444301427.ch15.
I Considering Pute strategy NE (PSNE) for model of three-players with an owner, a strong bidder and a regular bidder.
Player 1 is not to wander from [(S; H; H), (E – S; P – E; S – E)] to [(No S; H; H), (E; P – E; -E)]
Player 2 is not to wander from [(S; H; H), (E – S; P – E; S – E)] to [(S; A; H), (E/2 – S; P/2 + S/2; P/2 + S/2 – E)]
Player 3 is not to wander from [(S; H; H), (E – S; P – E; S – E)] to [(S; H; A), (E – S; P – E; S)]
Player 1 is not to wander from [(No S; A; A), (0; P; 0)] to [(S; A; A), (-S; P; S)]
Player 2 is not to wander from [(No S; A; A), (0; P; 0)] to [(No S; H; A), (E; P –
Player 3 is not to wander from [(No S; A; A), (0; P; 0)] to [(No S; A; H), (E/2; P/2; P/2 –E/2)]
Player 1 is not to wander from [(S; A; A), (-S; P; S)] to [(No S; A; A), (0; P; 0)]
Player 2 is not to wander from [(S; A; A), (-S; P; S)] to [(S; H; A), (E – S; P – E; S)]
Player 3 is not to wander from [(S; A; A), (-S; P; S)] to [(S; A; H), (E/2; P/2 + S/2; P/2 + S/2 – E)]
Player 1 is not to wander from [(S; H; A), (E – S; P – E; S)] to [(No S; H; A), (E;
Player 2 is not to wander from [(S; H; A), (E – S; P – E; S)] to [(S; A; A), (-S; P; S)]
Player 3 is not to wander from [(S; H; A), (E – S; P – E; S)] to [(S; H; H), (E – S; P – E; S – E)]
Player 1 is not to wander from [(S; A; H), (E/2 – S; P/2 + S/2; P/2 + S/2 – E)] to [(No S; A; H), (E/2; P/2; P/2 –E/2)]
Player 2 is not to wander from [(S; A; H), (E/2 – S; P/2 + S/2; P/2 + S/2 – E)] to [(S; H; H), (E – S; P – E; S – E)]
Player 3 is not to wander from [(S; A; H), (E/2 – S; P/2 + S/2; P/2 + S/2 – E)] to [(S; A; A), (-S; P; S)]
Player 1 is not to wander from [(No S; H; H), (E; P –E; -E)] to [(S; H; H), (E – S; P – E; S – E)]
Player 2 is not to wander from [(No S; H; H), (E; P –E; -E)] to [(No S; A; H), (ןE/2; P/2; P/2 –ןE/2)]
Player 3 is not to wander from [(No S; H; H), (E; P –E; -E)] to [(No S; H; A), (E; P – E; 0)]
Player 1 is not to wander from [(No S; H; A), (E; P –E; 0)] to [(S; H; A), (E – S;
Player 2 is not to wander from [(No S; H; A), (E; P –E; 0)] to [(No S; A; A), (0; P; 0)]
Player 3 is not to wander from [(No S; H; A), (E; P –E; 0)] to [(No S; H; H), (E; P –E; -E)]
Player 1 is not to wander from [(No S; A; H), (E/2; P/2; P/2 –E/2)] to [(S; A; H),(E/2 – S; P/2 + S/2; P/2 + S/2 – E)]
Player 2 is not to wander from [(No S; A; H), (E/2; P/2; P/2 –E/2)] to [(No S; H; H), (E; P –E; -E)]
Player 3 is not to wander from [(No S; A; H), (E/2; P/2; P/2 –E/2)] to [(No S; A; A), (0; P; 0)]
II Considering Pute strategy NE (PSNE) for model of four-players with the owner, two strong bidders, one regular bidder.
Player 1 is not to wander from [(S; 2H; H), (E – S; P/2 + S/2 – E; P/2 + S/2 – E; -E)] to [(No S; 3H), (E; P/2 –E; P/2 –E; -E)]
Player 2, 3 is not to wander from [(S; 2H; H), (E – S; P/2 + S/2 – E; P/2 + S/2 – E; - E)] to [(S; A; 2H), (E – S; S/2; P – E; S/2 – E)]
Player 4 is not to wander from [(S; 2H; H), (E – S; P/2 + S/2 – E; P/2 + S/2 – E; -E)] to [(S; 2H; A), (E – S; P/2 + S/2 – E; P/2 + S/2 – E; 0)]
Player 1 is not to wander from [(No S; 2A; A), (0; P/2; P/2; 0)] to [(S; 3A), (-S; P/2 + S/2; P/2 + S/2; 0)]
Player 2, 3 is not to wander from [(No S; 2A; A), (0; P/2; P/2; 0)] to [(No S; H; 2A), (ჴE; P –ჴE; 0; 0)]
Player 4 is not to wander from [(No S; 2A; A), (0; P/2; P/2; 0)] to [(No S; 2A; H), (ჴE/3; P/3; P/3; P/3 –ჴE)]
Player 1 is not to wander from [(S; 2H; A), (E – S; P/2 + S/2 – E; P/2 + S/2 – E; 0)] to [(No S; 2H; A), (ჴE; P/2 –ჴE; P/2 –ჴE; 0)]
Player 2, 3 is not to wander from [(S; 2H; A), (E – S; P/2 + S/2 – E; P/2 + S/2 – E; 0)] to [(S; A; H; A), (E – S; S; P – E; 0)]
Player 4 is not to wander from [(S; 2H; A), (E – S; P/2 + S/2 – E; P/2 + S/2 – E; 0)] to [(S; 3H), (E – S; P/2 + S/2 – E; P/2 + S/2 – E; -E)]
Player 1 is not to wander from [(No S; 2H; A), (ჴE; P/2 –ჴE; P/2 –ჴE; 0)] to [(S; 2H; A), (E – S; P/2 + S/2 – E; P/2 + S/2 – E; 0)] ჴE > E – S => S/(1 –ჴ) > E
Player 2, 3 is not to wander from [(No S; 2H; A), (ჴE; P/2 –ჴE; P/2 –ჴE; 0)] to [(No S; A; H; A), (ჴE; 0; P –ჴE; 0)]
Player 4 is not to wander from [(No S; 2H; A), (ჴE; P/2 –ჴE; P/2 –ჴE; 0)] to [(No S; 3H), (E; P/2 –E; P/2 –E; -E)]
Player 1 is not to wander from [(S; H; A; H), (E – S; P – E; S/2; S/2 – E)] to [(No S; H; A; H), (ჴE; P -ჴE; 0; -ჴE)]
Player 2 is not to wander from [(S; H; A; H), (E – S; P – E; S/2; S/2 – E)] to [(S; 2A; H), (E/3 – S; P/3 + S/3; P/3 + S/3; P/3 + S/3 - E)]
Player 3 is not to wander from [(S; H; A; H), (E – S; P – E; S/2; S/2 – E)] to [(S; 2H; H), (E – S; P/2 + S/2 – E; P/2 + S/2 – E; -E)]
Player 4 is not to wander from [(S; H; A; H), (E – S; P – E; S/2; S/2 – E)] to [(S; H; A; A), E – S; P – E; S; 0)]
Player 1 is not to wander from [(No S; H; A; H), (ჴE; P -ჴE; 0; -ჴE)] to [(S; H; A; H), (E – S; P – E; S/2; S/2 – E)] ჴE > E – S => S/(1 –ჴ) > E
Player 2 is not to wander from [(No S; H; A; H), (ჴE; P -ჴE; 0; -ჴE)] to [(No S; 2A; H), (ჴE/3; P/3; P/3; P/3 -ჴE)]
Player 3 is not to wander from [(No S; H; A; H), (ჴE; P -ჴE; 0; -ჴE)] to [(No S; 2H; H), (ჴE; P/2 -ჴE; P/2 -ჴE; -ჴE)]
Player 4 is not to wander from [(No S; H; A; H), (ჴE; P -ჴE; 0; -ჴE)] to [(No S; H; A; A), (ჴE; P - ჴE; 0; 0)]
[(No S; H; A; H), (ჴE; P -ჴE; 0; -ჴE)] and [(No S; A; H; H), (ჴE; 0; P -ჴE; -ჴE)] is not PSNE
Player 1 is not to wander from [(S; H; A; A), (E – S; P – E; S; 0)] to [(No S; H; A; A), (ჴE; P -ჴE; 0; 0)]
Player 2 is not to wander from [(S; H; A; A), (E – S; P – E; S; 0)] to [(S; 2A; A), (-S; P/2 + S/2; P/2 + S/2; 0)]
Player 3 is not to wander from [(S; H; A; A), (E – S; P – E; S; 0)] to [(S; 2H; A), (E – S; P/2 + S/2 – E; P/2 + S/2 – E; 0)]
Player 4 is not to wander from [(S; H; A; A), (E – S; P – E; S; 0)] to [(S; H; A; H), (E – S; P – E; S/2; S/2 – E)]
8 [(No S; H; A; A), (ჴE; P -ჴE; 0; 0)] and [(No S; A; H; A), (ჴE; 0; P -ჴE; 0)] Player 1 is not to wander from [(No S; H; A; A), (ჴE; P -ჴE; 0; 0)] to [(S; H; A; A), (E – S; P – E; S; 0)] ჴE > E – S => S/(1 –ჴ) > E
Player 2 is not to wander from [(No S; H; A; A), (ჴE; P -ჴE; 0; 0)] to [(No S; 2H; A), (ჴE; P/2 –ჴE; P/2 –ჴE; 0)]
Player 3 is not to wander from [(No S; H; A; A), (ჴE; P -ჴE; 0; 0)] to [(No S; 2A; A), (0; P/2; P/2; 0)]
Player 4 is not to wander from [(No S; H; A; A), (ჴE; P -ჴE; 0; 0)] to [(No S; H; A; H), (ჴE; P -ჴE; 0; -ჴE)]
[(No S; H; A; A), (ჴE; P -ჴE; 0; 0)] and [(No S; A; H; A), (ჴE; 0; P -ჴE; 0)]is not PSNE
Player 1 is not to wander from [(S; 2A; H), (E/3 – S; P/3 + S/3; P/3 + S/3; P/3 + S/3 – E)] to [(No S; 2A; H), (ჴE/3; P/3; P/3; P/3 -ჴE)]
Player 2, 3 is not to wander from [(S; 2A; H), (E/3 – S; P/3 + S/3; P/3 + S/3; P/3 + S/3 – E)] to [(S; H; A; H), (E – S; P – E; S/2; S/2 – E)]
Player 4 is not to wander from [(S; 2A; H), (E/3 – S; P/3 + S/3; P/3 + S/3; P/3 + S/3 – E)] to [(S; 2A; A), (-S; P/2 + S/2; P/2 + S/2; 0)]
[(S; 2A; H), (E/3 – S; P/3 + S/3; P/3 + S/3; P/3 + S/3 – E)]is PSNE when E > max[(3S/(1 –ჴ); (2P – S)/3] and E < (P + S)/3
Player 1 is not to wander from [(S; 2A; A), (-S; P/2 + S/2; P/2 + S/2; 0)] to [(No S; 2A; A), (0; P/2; P/2; 0)]
Player 2, 3 is not to wander from [(S; 2A; A), (-S; P/2 + S/2; P/2 + S/2; 0)] to [(S; H; A; A), (E – S; P – E; S; 0)]
Player 4 is not to wander from [(S; 2A; A), (-S; P/2 + S/2; P/2 + S/2; 0)] to [(S; 2A; H), (E/3 – S; P/3 + S/3; P/3 + S/3; P/3 + S/3 – E)]
Player 1 is not to wander from [(No S; 2A; H), (ჴE/3; P/3; P/3; P/3 -ჴE)] to [(S; 2A; H), (E/3 – S; P/3 + S/3; P/3 + S/3; P/3 + S/3 – E)] ჴE/3 > E/3 – S => 3S/(1 –ჴ) > E
Player 2, 3 is not to wander from [(No S; 2A; H), (ჴE/3; P/3; P/3; P/3 -ჴE)] to [(No S; H; A; H), (ჴE; P -ჴE; 0; -ჴE)]
Player 4 is not to wander from [(No S; 2A; H), (ჴE/3; P/3; P/3; P/3 -ჴE)] to [(No S; 2A; A), (0; P/2; P/2; 0)]
Player 1 is not to wander from [(No S; 2H; H), (ჴE; P/2 -ჴE; P/2 -ჴE; -ჴE)] to [(S; 2H; H), (E – S; P/2 + S/2 – E; P/2 + S/2 – E; -E)] ჴE > E – S => S/(1 –ჴ) > E
Player 2, 3 is not to wander from [(No S; 2H; H), (ჴE; P/2 -ჴE; P/2 -ჴE; -ჴE)] to [(No S; A; H; H), (ჴE; 0; P -ჴE; -ჴE)]
Player 4 is not to wander from [(No S; 2H; H), (ჴE; P/2 -ჴE; P/2 -ჴE; -ჴE)] to [(No S; 2H; A), ((ჴE; P/2 –ჴE; P/2 –ჴE; 0)]
III Considering Pure strategy NE (PSNE) for model of four-players with the owner, one strong bidder, two regular bidder.
Player 1 is not to wander from [(S; H; 2H), (E – S; P – E; S/2 – E; S/2 – E)] to [(No S; H; 2H), (ჴE; P –ჴE; -ჴE; -ჴE)] Ȃ ჴ ֜ Ȁ ൏ ͳȂჴ
Player 2 is not to wander from [(S; H; 2H), (E – S; P – E; S/2 – E; S/2 – E)] to [(S; A; 2H), (2E/3 – S; P/3 + S/3; P/3 + S/3 – E; P/3 + S/3 – E)] Ȃ Ȁ͵ Ȁ͵ ֜ ʹܲ െ ܵ Ȁ͵ ܧ
Player 3, 4 is not to wander from [(S; H; 2H), (E – S; P – E; S/2 – E; S/2 – E)] to [(S; H; A; H), (E – S; P – E; 0; S – E)] Ȁʹ Ȃ Ͳ ֜ ʹ
Player 1 is not to wander from [(No S; A; 2A), (0; P; 0; 0)] to [(S; A; 2A), (-S; P; S/2; S/2)] Ͳ െ ֜
Player 2 is not to wander from [(No S; A; 2A), (0; P; 0; 0)] to [(No S; H; 2A), (ჴE; P
Player 3, 4 is not to wander from [(No S; A; 2A), (0; P; 0; 0)] to [(No S; H; A; H), (ჴE; P –ჴE; 0; -ჴE)] Ͳ Ͳ ֜
Player 1 is not to wander from [(S; H; 2A), (E – S; P – E; S/2; S/2)] to [(No S; H;2A), (ჴE; P -ჴE; 0; 0)] െ ჴ ֜ Ȁ ൏ ͳȂჴ
Player 2 is not to wander from [(S; H; 2A), (E – S; P – E; S/2; S/2)] to [(S; A; 2A), (- S; P; S/2; S/2)] െ ֜
Player 3, 4 is not to wander from [(S; H; 2A), (E – S; P – E; S/2; S/2)] to [(S; H; A; H), (E – S; P – E; 0; S – E)] Ȁʹ Ͳ ֜
Player 1 is not to wander from [(No S; H; 2A), (ჴE; P - ჴE; 0; 0)] to [(S; H; 2A), (E – S; P – E; S/2; S/2)] ჴ െ ֜ Ȁ ͳȂჴ
Player 2 is not to wander from [(No S; H; 2A), (ჴE; P -ჴE; 0; 0)] to [(No S; A; 2A), (0; P; 0; 0)] െ ჴ ֜
Player 3, 4 is not to wander from [(No S; H; 2A), (ჴE; P -ჴE; 0; 0)] to [(No S; H; A; H), (ჴE; P –ჴE; 0; -ჴE)] Ͳ Ͳ ֜
Player 1 is not to wander from [(S; A; 2H), (2E/3 – S; P/3 + S/3; P/3 + S/3 – E; P/3 + S/3 – E)] to [(No S; A; 2H), (2ჴE/3; P/3; P/3 –ჴE; P/3 –ჴE)] ʹȀ͵ Ȃ ʹჴȀ͵ ֜ ͵Ȁ ʹ ͳȂჴ
Player 2 is not to wander from [(S; A; 2H), (2E/3 – S; P/3 + S/3; P/3 + S/3 – E; P/3 + S/3 – E)] to [(S; H; 2H), (E – S; P – E; S/2 – E; S/2 – E)] Ȁ͵ Ȁ͵ Ȃ ֜ ൏ ʹܲ െ ܵ Ȁ͵
Player 3, 4 is not to wander from [(S; A; 2H), (2E/3 – S; P/3 + S/3; P/3 + S/3 – E; P/3 + S/3 – E)] to [(S; A; A; H), (E/2 – S; P/2 + S/2; 0; P/2 + S/2 – E)] Ȁ͵ Ȁ͵Ȃ Ͳ ֜ ൏ ܲ ܵ Ȁ͵
Player 1 is not to wander from [(No S; A; 2H), (2ჴE/3; P/3; P/3 –ჴE; P/3 –ჴE)] to [(S; A; 2H), (2E/3 – S; P/3 + S/3; P/3 + S/3 – E; P/3 + S/3 – E)] ʹჴȀ͵ ʹȀ͵Ȃ ֜ ൏ ͵Ȁ ʹ ͳȂჴ
Player 2 is not to wander from [(No S; A; 2H), (2ჴE/3; P/3; P/3 –ჴE; P/3 –ჴE)] to [(No S; H; 2H), (ჴE; P –ჴE; -ჴE; -ჴE)] Ȁ͵ Ȃჴ ֜ Ȁ ൏ ͵ჴȀʹ ֜
Player 3, 4 is not to wander from [(No S; A; 2H), (2ჴE/3; P/3; P/3 –ჴE; P/3 – ჴE)] to [(No S; A; H; A), (ჴE/2; P/2; P/2 –ჴE; 0)] Ȁ͵ Ȃ ჴ Ȁʹ Ȃ ჴ ֜
7 [(S; H; A; H), (E – S; P – E; 0; S – E)] or [(S; H; H; A), (E – S; P – E; S – E; 0)] Player 1 is not to wander from [(S; H; A; H), (E – S; P – E; 0; S – E)] to [(No S; H; A; H), (ჴE; P –ჴE; 0; -ჴE)] െ ჴ ֜ Ȁ ͳȂჴ ൏ ܧ
Player 2 is not to wander from [(S; H; A; H), (E – S; P – E; 0; S – E)] to [(S; A; A; H), (E/2 – S; P/2 + S/2; 0; P/2 + S/2 – E)] െ Ȁʹ Ȁʹ ֜ ܲ െ ܵ Ȁʹ
Player 3 is not to wander from [(S; H; A; H), (E – S; P – E; 0; S – E)] to [(S; H; H; H), (E – S; P – E; S/2 – E; S/2 – E)] Ͳ Ȁʹ Ȃ ֜ Ȁʹ
Player 4 is not to wander from [(S; H; A; H), (E – S; P – E; 0; S – E)] to [(S; H; A; A), (E – S; P – E; S/2; S/2)] Ȃ Ȁʹ ֜ ൏ Ȁʹ
Player 1 is not to wander from [(No S; H; A; H), (ჴE; P –ჴE; 0; -ჴE)] to [(S; H; A; H), (E – S; P – E; 0; S – E)] ჴ െ ֜ Ȁ ͳȂჴ
Player 2 is not to wander from [(No S; H; A; H), (ჴE; P – ჴE; 0; -ჴE)] to [(No S; A; A; H), (ჴE/2; P/2; 0; P/2 –ჴE)] Ȃ ჴ Ȁʹ ֜ Ȁʹჴ
Player 3 is not to wander from [(No S; H; A; H), (ჴE; P – ჴE; 0; -ჴE)] to [(No S; H; H; H), (ჴE; P –ჴE; -ჴE; -ჴE)] Ͳ െ ჴ ֜
Player 4 is not to wander from [(No S; H; A; H), (ჴE; P – ჴE; 0; -ჴE)] to [(No S; H; A; A), (E – S; P – E; S/2; S/2)] െჴ Ȁʹ ֜
[(No S; H; A; H), (ჴE; P –ჴE; 0; -ჴE)] and [(No S; H; H; A), (ჴE; P –ჴE; -ჴE; 0)] arenot PSNE.
Player 1 is not to wander from [(S; A; A; H), (E/2 – S; P/2 + S/2; 0; P/2 + S/2 – E)] to [(No S; A; A; H), (ჴE/2; P/2; 0; P/2 –ჴE)] Ȁʹ െ ჴȀʹ ֜ ʹȀ ൏ ͳȂჴ
Player 2 is not to wander from [(S; A; A; H), (E/2 – S; P/2 + S/2; 0; P/2 + S/2 – E)] to [(S; H; A; H), (E – S; P – E; 0; S – E)] Ȁʹ Ȁʹ െ ֜ ܲ െ ܵ Ȁʹ ൏
Player 3 is not to wander from [(S; A; A; H), (E/2 – S; P/2 + S/2; 0; P/2 + S/2 – E)] to [(S; A; H; H), (2E/3 – S; P/3 + S/3; P/3 + S/3 – E; P/3 + S/3 – E)] Ͳ Ȁ͵ Ȁ͵ Ȃ ֜
Player 4 is not to wander from [(S; A; A; H), (E/2 – S; P/2 + S/2; 0; P/2 + S/2 – E)] to [(S; A; A; A), (-S; P; S/2; S/2)] Ȁʹ Ȁʹ Ȃ Ȁʹ ֜ Ȁʹ
Player 1 is not to wander from [(No S; A; A; H), (ჴE/2; P/2; 0; P/2 –ჴE)] to [(S; A; A; H), (E/2 – S; P/2 + S/2; 0; P/2 + S/2 – E)] ჴȀʹ Ȁʹ െ ֜ ʹȀ ͳȂჴ
Player 2 is not to wander from [(No S; A; A; H), (ჴE/2; P/2; 0; P/2 –ჴE)] to [(No S; H; A; H), (ჴE; P –ჴE; 0; -ჴE)] Ȁʹ Ȃ ჴ ֜ ჴ Ȁʹ ֜
Player 3 is not to wander from [(No S; A; A; H), (ჴE/2; P/2; 0; P/2 –ჴE)] to [(No S; A; H; H), (2ჴE/3; P/3; P/3 – ჴE; P/3 –ჴE)] Ͳ Ȁ͵ Ȃ ჴ ֜ ჴ Ȁ͵ ֜
Player 4 is not to wander from [(No S; A; A; H), (ჴE/2; P/2; 0; P/2 –ჴE)] to [(No S; A; A; A), (0; P; 0; 0)] Ȁʹ Ȃ ჴ Ͳ ֜ Ȁʹ ჴ
[(No S; A; A; H), (ჴE/2; P/2; 0; P/2 – ჴE)] and [(No S; A; H; A), (ჴE/2; P/2; P/2 – ჴE; 0)] arenot PSNE.
Player 1 is not to wander from [(No S; H; 2H), (ჴE; P – ჴE; -ჴE; -ჴE)] to [(S; H; 2H), (E – S; P – E; S/2 – E; S/2 – E)] ჴ െ ֜ Ȁ ͳȂჴ
Player 2 is not to wander from [(No S; H; 2H), (ჴE; P –ჴE; -ჴE; -ჴE)] to [(No S; A; 2H), (2ჴE/3; P/3; P/3 – ჴE; P/3 –ჴE)] Ȃ ჴ Ȁ͵ ֜ ʹȀ͵ ჴ ֜
Player 3, 4 is not to wander from [(No S; H; 2H), (ჴE; P – ჴE; -ჴE; -ჴE)] to [(No S; H; A; H), (ჴE; P –ჴE; 0; -ჴE)] െჴ Ͳ ֜
Player 1 is not to wander from [(S; A; 2A), (-S; P; S/2; S/2)] to [(No S; A; 2A), (0; P; 0; 0)] െ Ͳ ֜
Player 2 is not to wander from [(S; A; 2A), (-S; P; S/2; S/2)] to [(S; H; 2A), (E – S; P – E; S/2; S/2)] െ ֜
Player 3, 4 is not to wander from [(S; A; 2A), (-S; P; S/2; S/2)] to [(S; A; H; A), (E/2 – S; P/2 + S/2; P/2 + S/2 – E; 0)] Ȁʹ Ȁʹ ȀʹȂ ֜ Ȁʹ