Bidding strategies in the Swedish housing marketumu.diva-portal.org/smash/get/diva2:1093030/FULLTEXT01.pdf · game will therefore depend on the players’ choices of strategic or

Bidding strategies inthe Swedish housing market

Malin Norling Sofia Hjulfors

Autumn semester 2016Bachelor thesis, 15 hpBachelor degree of mathematics, 180 hpDepartment of mathematics and mathematical statistics

Acknowledgements

We both would like to thank the real estate agencies and the people whohelped us and provided bidding lists that we could analyze;Fastighetsbyran - Molnlycke,Fastighetsbyran - Bromma,Fastighetsbyran - Kungsangen,Maklarhuset - Askim.We would specially like to thank Thomas Hansson from Maklarhuset, Askim,who we got the opportunity to interview to get a bigger understanding ofthe housing market and the strategies involved.

I would like to dedicate this thesis to my loving father who passed away 2014.He always encouraged me to push my self and to believe in my self. He wasand still is my role model. I also would like to thank my mom Anett, mysister Emelie, and my boyfriend Fredrik, for their love and support to me.- Malin Norling

I am thankful for the support I got during this thesis process from friendsand family. I would specially like to thank my father, Stefan, for pushing usboth during this period, to inspire us to think outside the box. I would liketo thank Pinja, may she rest in peace, that took us on inspiring walks whenwe needed breaks.- Sofia Hjulfors

Abstract

This report focuses on an introduction of game theoretical modelsand how they can be applied in the Swedish housing market. Gametheory is a study of mathematical models of human conflicts and co-operation between rational decision makers within a competitive sit-uation. There are several different strategies that a player can use.In this thesis each strategy is assigned to one player. So how will theplayers behave in a game, and what strategy is the most successful?By using the software MatLab, the authors creates a game where thestrategies assigned to each player gets randomly distributed budgetsand are randomly selected to place bids during the game. The game isthen played 1 000 000 times to see what strategy is the most success-ful. It is also tested to see what strategy is the most successful if theplayers have the same budgets. The authors conclude that in practiceit is the size of the budget that determines who will win the bidding,hence there are minor differences between the different strategies inhow much they pay on average to win.

Sammanfattning

Denna rapport fokuserar pa en introduktion av spelteoretiska mo-deller och hur de kan som kopplas till den svenska bostadsmarknaden.Spelteori ar en studie om matematiska modeller for manskliga konflik-ter och samarbete mellan rationella beslutstagare i en konkurrensut-satt marknad. Det finns flera olika strategier en spelare kan anvandasig av. I denna rapport blir varje spelare tilldelad en strategi. Sa fraganstalls hur spelarna kommer att bete sig och vilken av strategiernasom ar den mest framgangsrika. Genom att anvanda programvaranMatLab, skapar forfattarna ett program dar varje strategi ar tilldeladvarje spelare och som helt slumpmassigt far en budget och aven blirslumpmassigt valda att spela, d.v.s lagga bud under spelets gang. Spe-let spelas darefter 1 000 000 ganger for att se vilken av strategiernasom ar mest framgangsrik. Det ar aven testat att se vilken strategisom far bast resultat om de alla har samma budget. Forfattarna drarslutsatsen att i praktiken ar det storleken pa budgeten som bestammervem som vinner budgivningen, dock att det aven finns mindre skillna-der mellan strategierna som bestammer hur mycket de i genomsnittfar betala nar de vinner.

”You need to learn the rules of the game,and then you have to play better than anyone else.”

- Albert Einstein

Contents

1 Introduction 1

2 Game theory 32.1 Rules of the game . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Cooperative and non-cooperative games . . . . . . . . 52.1.2 Simultaneous and sequential games . . . . . . . . . . . 6

2.2 Normal form games . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Extensive form games . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 Extensive games with perfect information . . . . . . . 82.4.2 Extensive games with imperfect information . . . . . . 10

2.5 Bayesian games . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 Evolutionary game theory . . . . . . . . . . . . . . . . . . . . 122.7 Auction theory . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 The housing market 143.1 Interview with a real estate agent . . . . . . . . . . . . . . . . 143.2 Auction rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Bidding strategies . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 The auction game 184.1 Rules of the game . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 The game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Results and analysis of the game 20

6 Conclusion 24

A Appendix: MATLAB codes 28A.1 The auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28A.2 Bid counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31A.3 Competitors left . . . . . . . . . . . . . . . . . . . . . . . . . . 32A.4 Winner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32A.5 Average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32A.6 Strategy 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33A.7 Strategy 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34A.8 Strategy 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35A.9 Strategy 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37A.10 Strategy 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38A.11 Strategy 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

List of Figures

1 The Sharing game . . . . . . . . . . . . . . . . . . . . . . . . . 92 Randomly distributed budgets . . . . . . . . . . . . . . . . . . 203 Fixed budgets . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

List of Tables

1 Average price paid from randomly played game . . . . . . . . 212 Bidding list with random budgets . . . . . . . . . . . . . . . . 23

Nomenclature

∈ Element of∪ Union∩ Intersection∅ Empty set× Cartesian productR Real numbersG Gamei Playern Number of playersN Set of all players, N =

{1, ..., n

}Si Set of pure strategies for each player i ∈ N , Si =

{s1, ..., sm

}S∗i Set of mixed strategies for each player i ∈ N , S∗i =

{s∗1, ..., s

∗n

}S Set of all strategic profiles, S = S1 × ...× Sn

Ai Set of possible actions for each player i ∈ N , Ai ={a1, ..., an

}I Set of information, I =

{I1, ..., In

}Θi Set of possible types for player i ∈ N , Θi =

{θ1, ..., θn

}H Set of non-terminal nodesZ Set of terminal nodes, Z ∩H = ∅P Probability vector, P = (p1, ..., pm)U Payoff (utility)α Action functionρ Player functionσ Successor function

1 Introduction

Game theory is a mathematical tool that studies the analysis of solving strate-gic problems of interaction among decision-makers and was originally intro-duced by von Neumann and Morgenstern in 1944 [15]. Each game has a setof rules and involves one or more decision-makers (players), whose actionsand moves affect both the player itself and the others [16]. The result of thegame will therefore depend on the players’ choices of strategic or randommoves depending on information, preferences, possibilities, and reactions.There are a variety of solution concepts in game theory, for example theNash equilibrium, who is arguably the most well-known to use to analyzepossible outcomes. Game theory is broadly used in a numerous of fields ineconomics, in everyday life, politics, and other game-related situations [3].The assumption of rational players is used in game theory [14]. Assumingeach player is seeking maximum utility and that all other players are doingthe same is essential for the logical predication of the game.

Auctions are a particular type of game that are studied in a branch of gametheory, which is known as auction theory. In many real-life situations we canfind auctions and they are often found when dealing with economic trans-actions. Auction theory is therefore an important tool for understandinginteraction among sellers and buyers. There are many different sets of rulesfor an auction that defines what type of auction it is. There is for examplethe sealed-bid auction of first and second price, the descending-price auc-tion, and the Japanese auction [17]. In this thesis, the authors will focus onthe auction model for open ascending bids, which is the model used in theSwedish housing market.

A potential value of applying auction theory to the housing market is togive real estate agents a better understanding of how the market work andalso the possibilities to analyze consequences of possible rule changes. It isalso important for the buyers in the housing market to understand how to getan object to a cheaper price by a well chosen strategy. There are, however,difficulties and challenges when applying auction theory in practical and real-life situations. There are unique circumstances for each auction, certain rulescan lead to changes and some factors goes outside the usual auction theory.Such factors could involve bids that are placed before the start of the auction,the psychology and mind set of each bidder and need of reactive strategies etc.

Despite this untapped potential, it thus remains unclear to what extent auc-tion theory applies. How should a game-theoretical model for the Swedish

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housing market be constructed?

The aim of this thesis is to give an introduction to game theory, with focuson the mathematical theory behind auctions, and also to research auctionsin the Swedish housing market. The purpose is to see how the auctions arerelated to games and to study effectiveness of selected strategies used in thehousing market.

The problem formulation for this thesis has been to see if there is a gametheoretical model for how rational players are behaving in a bidding game,which are represented by the rules that is in the Swedish housing market.

The methodology was to see how game theory is a study of mathematicalmodels of human conflict and cooperation between rational decision makerswithin a competitive situation. When implementing it into the Swedish hous-ing market and the bidding game, the authors used the game theory methods;extensive form, perfect and incomplete information, mixed and pure strate-gies, non-cooperative, and n-player. The authors chose these methods toshow the connection related to the Swedish housing market, furthermore anintroduction of evolutionary game theory and auction theory, which are allexplained in section 2. When constructing the game, the authors needed tofind all different strategies that bidders can use in auctions in the Swedishhousing market, as well as the certain rules that the bidders need to follow.The strategies and the rules are explained in section 3. Section 4 explainsthe author’s game that is implemented in MatLab.

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2 Game theory

This chapter will give a brief introduction to some of game theory’s conceptsthat have been used by the authors to build the auction game.

2.1 Rules of the game

A set of rules describes the game-theoretical model. The essential elementsconsist of players, information, actions and payoff which all depends on theparticular game that is played, that usually is of normal form or extensiveform. This general description of the elements and rules will be used for eval-uating our own game-theoretical model in the housing market [16], includingsome brief introduction of other types of game theoretical models.

Players

The players, denoted N , consist of the number of people that are playingthe game. There are games that can be played consisting of only one player,such as roulette, but more often there are more than one player in the game.These players will take decisions in the form of actions with the goal tomaximize its utility. Each player knows that the other players in the gamealso have the goal to maximize its utility [15], though the utility might beof different value among the players. Moreover, the players’ interests couldboth be matching or conflicting, meaning that the players can cooperate ornot to reach utility.

Information

The rules of the game decide what kind of information about different vari-ables is available to the players [3]. The information, denoted I, can alsochange during the game. Games where perfect information occur give allthe players the same information throughout the game [3]. The players haveknowledge about the other players moves like in a game of chess where all thechess pieces are visual for both players throughout the game. In contrary,imperfect information has the opposite effect. The players cannot see theother players moves.

The information available in the game is asymmetric or symmetric. In sym-metric information all the players have the same information, which meansthat the players know about the other players moves and payoffs at everynode and they will then base their own strategies on that information [3].

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In the asymmetric information, all the players have different informationconcerning the other players moves and payoffs. For example that player 1knows some information that player 2 does not.

How well the players know the structure depends on if the game has completeor incomplete information [14]. Given a game with complete information, theplayers are aware of the structure of the game. The players are aware of howthe game is played, all possible actions and the payoffs for those outcomes.A game with incomplete information is more complex since there are manyvariables unknown. The players might be unaware of how many players thereare in the game, what their payoffs are and what actions are possible. Theyare also unaware of how the outcomes of the moves will affect the game [9],what other players know and if they have the same information as otherplayers.

Actions

The actions, denoted A, are the moves that the players will do depending onprevious actions from other players, information available, preferences, rulesetc. Each game has a number of rules to follow to be able to make actions inthe game. Rules might include perfect or imperfect information, availabilityor not with communication among players, money to enter the game or not.Seeing that players are rational, the actions would reflect on that assumption,though in reality people also make irrational or random decisions [3].

Payoff

The players’ payoff, denoted U , is the utility that the players receive whenthe game has finished [3]. This depends on the strategies chosen duringthe game, the actions made by other players and other variables tied tothe specific game. Predicting the payoff in a real life situation is almostimpossible due to the possibility of irrational players. In a game-theoreticalmodel it is possible to predict the players’ possible outcomes and expectedutility when it exists perfect and complete information due to the sharedinformation between the players.

Mixed and pure strategies

A game consist of players taking actions and making strategies throughoutthe game. Actions of other players can influence the strategy or be of ran-domized type, giving two types of strategy sets, pure and mixed [6]. In a

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game with pure strategy the player decides a strategy profile to use through-out the game. This profile has rules for every possible action made of othersin the game, making the action process fixed for every node in the game. Ina game with mixed strategy the player choose a probability distribution overthe set of pure strategies and the strategies become randomized rather thanfixed [18].

Definition 1. If a player, i has a set of Si pure strategies {s1,...,sm}, a mixedstrategy set, S∗i is defined by the probability vector, P = (p1, ..., pm), wherethe players’ strategy will be selected. Observe that for S∗i to be well defined,the sum of the probabilities should equal 1 and each of the probabilities(p1, ..., pm) must lie between 0 and 1. Observe that when choosing a mixedstrategy (p1 = 0, ..., p2 = 1, ..., pm = 0), it is equivalent to choosing the purestrategy set S2 [6].

2.1.1 Cooperative and non-cooperative games

Game theory consists of two branches, cooperative and non-cooperative gametheory [1]. To understand these two branches it is important to know that’cooperative’ and ’non-cooperative’ are technical forms and do not meancollaboration between the players in the model. The cooperative game canmodel extreme competition as much as the non-cooperative game and thenon-cooperative game can model collaboration [1].

Cooperative game theory

The cooperative game theory is a model that states what payoffs each groupof players, or coalition, obtains in the cooperation of the players [1]. It is notdefinitive that the process forms the coalition, which can be when the playersare in different groups in an parliament [18]. Every player in the group havedifferent strengths, which is based on the number of players in the group.The groups can form a majority based on the coalition, but need to processa negotiation in any agreement that a vote is achieved [18].

Non-cooperative game theory

The non-cooperative game theory is a model of situations in which playersare involved in an interactive process whose outcome is determined by eachplayers’ decision [7]. The players are not allowed to make binding agreementswith other players. They can communicate with each other and discuss

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united strategies of actions to the game, but during the game the players arenot allowed to communicate to each other and are then autonomous decisionmakers [7]. The theory consists of three main subjects. The first one is theprogress of technical models that build frame works for the representationof the game. The second is the concept that engages the idea of rationalbehaviour, and the third and last is the mathematical tools that is used inorder to demonstrate statements of the concepts of the existence and theequilibrium [7].

2.1.2 Simultaneous and sequential games

The models in game theory states that every action has a reaction, and thatthey follow in two ways: sequential and simultaneously.

Sequential game theory

In the sequential game theory the players take interchanging turns to maketheir choices. In sequential games it is important to know who is going tomake the first move [3], since it can be both an advantage and an disadvantageto start. Depending on the game structure and information, the playersmight know the actions taken in the previous turns and be able to observeeach other and follow the other players future moves. If this is true, theplayer can use the information to estimate their own upcoming move, andthis is called the “backward induction”. An example for this is the gamechess. When making the move, the player knows the previous moves of theother player and can use that to predict the next move. In that way, theplayer can use that information for his own strategies and actions.

Simultaneous game theory

In the simultaneous game theory the players make their decision at the sametime [3]. As long as they do not know about the other players’ choices inaction, they do not need to make their decisions at the same time. It canbe very difficult in this type of game since the players do not know what theother players strategies are.

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2.2 Normal form games

The normal form game, also called the strategic form game, is a game G witha finite set of N players [13]. The game is played simultaneous at a one-timedecision without any communication with other players. For every player i,they have their own set of actions Ai, and a preference relation on the set ofaction profiles.

Definition 2. A normal form game is a tuple 〈N,A, u〉 that consists of [17]

• A finite set N of n players, indexed by i

• A = A1× ...×An. For each player i ∈ N a non-empty set Ai of actionsavailable to i

• u = (u1, ..., un) where ui : A → R is a real-valued utility (or payoff)function for player i

2.3 Nash equilibrium

In 1951 John Nash developed game theory tools and introduced the conceptof the Nash equilibrium of a strategic game. It assumes that each player’sstrategy maximizes his payoff [3]. The Nash equilibrium, also called thestrategic equilibrium, captures therefore a steady state of the play of the game[13]. All players expect that the other players will follow the strategy thatmaximizes his payoff and that they cannot change their strategy unilaterallyand get a better payoff. By separating the strategy of player i from all otherplayers, it will be denoted by s−i.

Definition 3. Strategy profile s∗ is a Nash equilibrium if, for all i = 1, ..., Nand all si ∈ Si, ui(s

∗i , s∗−i) ≥ ui(si, s

∗−i) [12]

The Nash equilibrium can be divided into two categories, weak and strict,which depends on if the players’ strategies is a unique best response to theother players’ strategies. The weak Nash equilibrium is less stable than thestrict, since at least one player has a best response to the other players’strategies,which is not his equilibrium strategy. Depending on the game ofpure strategy the Nash equilibrium can be either weak or strict, and in thegame of mixed strategy the Nash equilibrium is weak [17].

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2.4 Extensive form games

Games in extensive form, introduced by John Von Neumann involves differentstages called nodes, that can be presented in a tree graph [18]. At every nodeh the player can choose a strategy with knowledge from some informationabout the previous moves by other players. For this reason, the nodes canbe seen as its history to follow all the choices from the root node to h [17].Furthermore, the player’s moves at every node are not made simultaneously,but instead sequentially with an order made by the rules of the game. Thestructure of the game is the information, what each player knows about theother players and their moves, and the utility for all possible outcomes. Everygame starts at the initial node and continues through the tree. Every non-terminal node belongs to one player. At that node, the player can choosebetween a number of different strategies, and each strategy leads to anothernode [18]. For games with perfect information, each node in the tree graphwill represent all possible strategies that can be chosen by a player. Theplayer can have a mixed or pure strategy profile, where the mixed strategyprofile is represented by a probability distribution over a pure strategy set.The players in the game have complete knowledge about the history of actionsof all players in the game and also at what node they are in [18]. Gameswith imperfect information have restricted information or no informationconcerning the previous actions and behaviour in the game and because ofthat, the players must make assumptions. The choice nodes are partitionedinto information sets. With that being defined, the player cannot distinguishbetween two choice nodes with the same information set [17].

2.4.1 Extensive games with perfect information

Definition 4. A finite perfect-information game in extensive form is a tupleG = (N,A,H,Z, α, ρ, σ, u) [17] that includes:

• A finite set N of n players, indexed by i

• A set of actions A

• A set of non-terminal choice nodes H

• A set of terminal choice nodes Z, where Z ∩H = ∅

• An action function α : H 7→ 2A which assigns to each choice node a setof possible actions

• A player function ρ : H 7→ N which assigns to each non-terminal nodea player i ∈ N who chooses an action at that node

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• A successor function σ : H × A 7→ H ∪ Z which maps a choice nodeand an action to a new choice node or terminal node such that for allh1, h2 ∈ H and a1, a2 ∈ A.If σ(h1, a1) = σ(h2, a2) then h1 = h2 and a1 = a2

• A real-valued utility function for player i on the terminal nodes Zu = (u1, ..., un) where ui : Z 7→ R

Figure 1: The Sharing game

Figure 1, the sharing game from previous page is a perfect-information gamein extensive form with pure strategies [17]. The game shows the pure strate-gies available when two siblings, one brother and one sister, follow a protocolof agreeing how to share two identical gifts which they value equally. First,the brother suggests a split, which can be divided into three; he keeps both,she keeps both or they keep one each. After the split, the sister decides if sheaccepts the split or not depending on the preferable outcome. If the sisterwould decide to not accept the brother’s split, they would both go emptyhanded. Looking at Figure 1, for the brother and sister to get gifts and beequally happy, the boy should suggest a split where they keep one gift eachand the sister should accept the split. The Sharing game is furthermore arepresentation of three pure strategies for player 1, the brother, and eightpure strategies for player 2, the sister [17].

S1 = {2-0, 1-1, 0-2}S2 = {(yes, yes, yes), (yes, yes, no), (yes, no, yes), (yes, no, no), (no, yes,yes), (no, yes, no), (no, no, yes), (no, no, no)}

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2.4.2 Extensive games with imperfect information

Definition 5. An imperfect-information game in extensive form is a tuple(N,A,H,Z, α, ρ, σ, u, I) [17], such that:

• (N,A,H,Z, α, ρ, σ, u) is a perfect-information game in extensive form.

• I = (I1, ..., In), where Ii = (Ii,1, ..., Ii,ki) is a set of equivalence classeson {h ∈ H : ρ(h) = i} with the property that α(h) = α(h′) andρ(h) = ρ(h′) whenever there exists a j for which h ∈ Ii,j and h′ ∈ Ii,j.

The definition tells us that for the player to not be able to distinguish thechoice nodes, the set of actions at each choice node in the information setmust be the same. If Ii,j ∈ Ii is an equivalence class, then we can use thenotation α(Ii,j) to describe all the actions available to player i at any nodein the information set Ii,j [17].

Definition 6. The imperfect-information game in extensive form is a tupleG = (N,A,H,Z, α, ρ, σ, u, I). Then the pure strategies of player i consist ofthe Cartesian product ΠIi,j∈Iiα(Ii,j) [17].

2.5 Bayesian games

Bayesian games introduce incomplete information in the game, where thestructure of the game is not fully known by the players. Preferences, strate-gies, budgets and utility functions might be some of the information that isnot shared among the players, making it more difficult to predict how theother players will act in the game and how to play against them [9]. Gameswith incomplete information were developed by Harsanyi (1967-1968) [9] whostated the differences in the private information among players as different”types” of players. Common knowledge was also introduced in the game,where each player represent a type of player and that the players only knowtheir own private type and not the others. Based on the players’ type theywill make beliefs and assumptions about other players’ type. They will alsomake assumptions about how other players will see and respond to the in-dividual player, their private utility function, the one of others and otherassumptions related to the game [9]. The Bayesian game can be defined indifferent ways depending on the underlying structure. Besides looking atdifferent type of players, Harsanyi found a way to reconstruct the concept

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of incomplete information into complete and imperfect information, withoutchanging the essential concept [9]. The information that is not shared amongthe players would be seen as though players have different information, i.e.imperfect information and that Nature plays a roll in the game leading toa hypothetical random first move [17]. Nature can be seen as a player witha mixed strategy profile to easier understand the concept. From here it ispossible to analyze the game and find a Bayesian-Nash equilibrium whichintuitively states that each player is doing his best to maximize his payoffgiven his beliefs and assumptions about the behavior of other players [17].Bayesian games are found in many real-life and economic situations wherethe information about utility functions is partly or not at all shared, such asin non-cooperative house auctions for example. In that situation, the biddersin the auction are solely aware about their own preferences and the utilityfunction regarding a certain house that they are bidding on, but not awareabout the other bidders’ preferences and utility functions that are bidding forthe same house. This makes in difficult for the bidders to make predictionsof the auction. Since bidders are not able to communicate with each other,the will instead make beliefs about how other bidders will put bids in theauction and how they should respond to succeed and win the house.

Definition 7. The Bayesian game reflecting types, where there is an un-certainty over the game’s utility function is a tuple (N,A,Θ, p, u) [17] thathas:

• A set N of n players, indexed by i

• A = A1× ...×An. For each player i ∈ N a non-empty set Ai of actionsavailable to i

• Θ = Θ1 × ...×Θn where Θi is the type space of player i

• p : Θ 7→ [0, 1] is common prior over types

• u = (u1, ..., un), where ui : A×Θ 7→ R

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2.6 Evolutionary game theory

Evolutionary game theory apply game theoretical ideas to model human andanimal behavior of strategic interaction. Some differences that divide evolu-tionary game theory from classical game theory is that the concept of ratio-nality is more relaxed, yet providing valid conclusions. It focuses more onthe dynamics of a population and natural selection, how individuals changeover time through a replicator rather than the individuals payoff maximiza-tion of a game and equilibrium [4]. The focal point of rationality has alsobeen proven by science in numerous testings that human actors are more of-ten irrational than not, making evolutionary game theory more applicable tostudy human behaviors and interactions than game theory [11]. The theorywas introduced and developed from evolutionary biology in 1930 [11] fromwhich they originally looked at how different species and their genes deter-mined characteristics of behaviours and how they evolve over time. Thesebehaviours got translated into strategic behaviors and the decisions madewhere seen as unconscious decisions, mutations or selections and thereforenot rational. Payoffs would be given when interactions among species oc-curred in a population and can therefore be seen as a game, where the fittestspecies would get the highest payoff by having a higher reproduction rateand drive out other species [2]. The strategic behaviors could be seen as abest response from a game theoretical perspective, but biologists were moreinterested in looking at possible predictions of stable strategies and the evo-lution of a population in isolation [15].

Evolutionary game theory has two different approaches for understandingand analyzing, one dynamic approach and one static. Both approaches as-sume large populations with random pairwise interaction among individuals.The dynamic approach is usually called the replicator dynamics [4]. It is amodel where individuals frequently interact in an population which give thempayoffs in terms of fitness. The main point of the replicator dynamics statesthat individuals make approximately copies of themselves by reproductionand that individuals with higher payoffs also reproduce faster than otherindividuals, making those individuals with a certain replicator-fitness to in-crease in the population. These replications can for example be a strategyin a game, cultural or a gene. Replications can also include random errorsas mutations [17]. The static approach is related to the replicator dynamicsand is called the evolutionarily stable strategy (ESS) which was introducedby Maynard Smith and Price [17]. Its stability concept states that if themajority of a population is using a certain mixed strategy, a small numberof entering individuals using another mutant strategy cannot successfully in-

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vade the population. The original strategy is therefore an evolutionary stablestrategy, which results in a higher payoff than the new mutant strategy, andwill therefore drive out the new individuals from the population. This isstrongly related to the Nash equilibrium in game theory. For larger numbersof individuals using a mutant strategy, they could be able to successfullyinvade the population [17].

2.7 Auction theory

Auctions are widely used for different economic transactions, computationalsettings, in war games and other real life situations. Auction theory hasbeen an important tool for understanding interactions between sellers andbuyers and moreover for understanding games of incomplete information ingame theory [8]. There are many kinds of auction-theoretical models forunderstanding resource allocation problems between self-interested agents.The sealed-bid auction of first and second price, the descending-price auctionand the Japanese auction are only some of the standard models used inauction theory. Introducing the auction type of open ascending bids, we willfocus on the auction model concerning the Swedish housing market in thisresearch [17].

Ascending auctions

In the ascending-bid auction, also known as the English auction, prices in-crease successively until there is only one bidder left that will pay the finalprice of the object. This is done either by sellers increasing the price thatthe bidders have to response to by accepting or leaving, by letting the bid-ders themselves increase the price in the open auction or by doing the sameprocedure electronically [8].

We consider auctions where the bidders have independent private values ofthe objects, meaning that each bidder knows how much they value the ob-ject, but has no information about the other bidders valuation in the auction,though signals from other bidders in the auction can affect the bidders’ val-uation of an object [8]. Due to these independent private values, it is morelikely that the bidder with the highest private value will win the object in anauction, like in sealed bid second-price auctions where the bidder bids onceand with his true value of the object [8], though, the signals from the otherbidders can also affect the winning outcome. Auctions where bidders have

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independent private values can be seen as games of incomplete informationin game theory.

3 The housing market

3.1 Interview with a real estate agent

To really understand how the housing market works we went to the realtorsMaklarhuset in Askim, Gothenburg and interviewed the real estate agentand partner of Maklarhuset, Thomas Hansson [5]. He informed us that theaverage of biddings on objects depends on the market conditions and whatkind of objects are for sale. He also stated that if it is an attractive objectthere can be up to 45-50 families that are interested but it also depends onthe value of the first bid. If the first bid is very high many speculators willdrop out, hence if it is low or on the asking price more speculators will jointhe auction. The time period of the auction is normally 1-2 days. When thereal estate agents evaluate the price of the object, they make a true pric-ing assumption on the object based on condition, other objects’ value in thearea today etc. Thomas Hanson made it clear that the objects’ value is theircorrect one when they get listed on the market, that it is not a lower priceto attract more speculators. He also pointed out that during periods whenthere are fewer object out on the market, the biddings tend to increase morethan expected. But what happens if the interest rate increases or decreases?When the interest rate increases or decreases it does not affect the market.The people that are selling their properties are not effected by the changesin interest rates. The only time a seller is potentially making a faster sellthan normal is when a divorce occur or the property owner has passed away.

In recent years it has become more common that the objects are sold be-fore the scheduled viewing of the object. The real estate agents are trying toprevent it, but if the seller finds the early bid good he will probably accepteven if no speculator has been there to see the property. Why this has be-come more common is probably that the buyer has earlier experienced thisform of process and missed out an object that has been sold to an early bid.For this reason and to not miss out on an object, more speculators makeearly bids. To place a bid before the viewing is one type of strategy. But isthere a strategy that is better than any other strategy? For all that Thomasknows, there is no strategy that is better than another. It all depends on theother buyers and the market at the specific time.

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The limiting knowledge for speculators about other speculators can be con-fusing for the involved. During the auction process the speculators mightthink that they are only four speculators if only four speculators have en-tered the auction with bids. Sometimes in the end of the auction one ormore speculators join. For the other speculators it seems as if they are newin the auction but the real estate agent has had knowledge about interestedspeculators from the start. Some speculators choose to wait with their firstbid until later in the auction to see the price development and patterns ofother speculators. When the auction is done and the seller has accepted thelast bid, the real estate agent saves all the bids for the auction and gives acomplete list to the seller and the buyer.

3.2 Auction rules

Auctions consist of two different models, open and closed bidding. Duringthe open bidding process, the speculator gives the real estate agent their bidon the object, who then announces it to the seller and the other speculators.The other speculators can then decide if they want to stop the auction ormake a new bid, which open possibilities for a bidding race between differ-ent speculators. During a closed bidding process, the speculators gets a timeframe when they can give their bid on the object, making the bidding processa one-shot opportunity. When the deadline is closed, the real estate agentgives the bids to the seller who then decides which bid and speculator is moreconvenient to sell to. The difference here from the open bidding process isthat the speculators don not know about the bids of the others. The spec-ulator has no obligation to get any information from the real estate agentabout the other speculators. When the winning speculator and the sellerwrites the contract, the real estate agent will give them the bidding list withinformation about the speculators and their bids. According to the Swedishwebsite maklarsamfundet.se, the most usual model is the open bidding, andwe will use this model in the research.

In the end of the auction process, the seller always decides who will winthe action. Even though it is normally the highest bidder that wins the auc-tion, there are cases where the seller chooses one of the other offers made.The seller could feel discomfort with the highest bidder, that the bidder doesnot seem reliable and serious about the agreements or that the bidder ismaking the sales process take longer time than needed.

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3.3 Bidding strategies

Looking at the housing market, there are several strategies to choose fromwhen it comes to bidding. During the auction event, it is possible to changethe strategy and the strategy chosen from the start might not be the actualstrategy played. Factors that influence the strategies are many. Some of thefactors include how many bidders there are in the auction and how they bid,what knowledge they have or previous experience from auctions and howmuch money the bidder is willing to spend depending on the budget andloan constraints from the bank. Getting excited and emotionally involvedwith early attachment to the object could also influence the bidding and thestrategy. This is a frequently occurring factor that is not rational from agame theoretical perspective.

Common strategies in the Swedish housing market

Asa Larsson, a real estate broker at the Swedish real estate agency Fastighets-byran, Lysekil, stated in the broker blog [10] some of the more commonstrategies used in the Swedish housing market.

Bidding early in the auction process. Bidding before the viewing ofthe object could potentially lead to a sell if the seller is eager to make a fastdeal. If not, it might also show both the seller and the broker that the bidderis very interested in buying the object, which can be valuable if the bidderstays until the end of the auction.

Start bidding late. Waiting with bidding until there is only one bidder leftshown in the auction could lead to a winning object by shocking the otherbidder. This strategy could potentially make the seller and the broker unsureif the bidder is reliable and serious.

Putting everything on one card. After deciding what maximum valueyou’re willing to pay for the object, you make a bid close to it. It is eitherall or nothing.

Waiting out the time. Before bidding, ask the broker if you can sleepon it over the weekend. With the extra time, the other opponents mightfinish other auctions instead, leaving fewer opponents in your auction. Thiscould potentially lead to a reversed effect, attracting new bidders to the auc-tion, or the seller is not willing to wait.

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Laying low. After a few rounds, start bidding lower and lower, makingthe auction potentially long and making the opponents tired. Making lowbids will also make it less risky to pay too much for the object.

Going out strong. Making high bids from the beginning and during theauction, could potentially scare opponents away.

Reacting quickly. Make a new bid as soon as there is movement in theauction. This shows all the parties involved that you’re interested and notleaving the auction.

Crossing borders. Many of the people interested in buying objects settheir highest budget constraints at even borders like 2 million, 2.5 millionetc. Bidding a little over the even borders could lead to a win.

Making irrational moves. People want to see patterns following theiropponents in the auction to possibly find a strategy against them. Being alittle bit irrational in the auction, choosing different strategies or no strategywill make the opponents confused.

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4 The auction game

In this section we have constructed an auction game in MatLab to presentdifferent strategies in the Swedish housing market. The auction game makesit possible to see how the different strategies could interact and if there is astrategy that is more efficient than others. We have found connections andapplied theory concepts from game theory to analyze this matter.

4.1 Rules of the game

From our interview with the real estate agent Thomas Hansson, the strategyinformation from Asa Larsson, and from our own researches we construct therules of our game.

There are 2 ≤ n ≤ 15 players in one game, and the game is of perfect-information and consists of players that are non-cooperative who have noknowledge about the other players in the game. The only thing they knowabout the other players is the number code that they have, and when theyare adding their bids. Moreover, the players know their own value of the ob-ject, but not the other players’ value, which makes it a game of incomplete-information. The goal for each player is to win the auction with the lowestpossible equilibrium. By using a certain strategy and bidding a value that ishigher than the current value and lower than the budget function, the playerswill aim to win the auction by having their bid as the last one in the auction.

The different strategies that we are using in the game are either pure ormixed. The game is played randomly and the bidding player will be selectedrandomly (no type of selected rotation between the players). The player canchoose to bid below, on, or above the opening price of the object. The playersin an on going game must bid above the current, existing bid, but the playercan also choose to wait until other players make a bid or else the player canchoose to exit the game. The auction ends when no player is bidding abovethe existing bid.

For each of the strategies there are individual rules that they are follow-ing. The rules of the strategies will be presented below, but for more detailsof the construction of the strategies, please see the appendix.

Strategy 1: Putting everything on one bidThe pure strategy is to give a bid that is 30 000 below his budget. If the bids

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after are still below the players budget, he will make a second bid which isthe value of his budget.

Strategy 2: Laying low The pure strategy is bidding either 5 000 or10 000. If the player makes it to the end and there is only one competitorleft, the player will bid every time a new bid is made until budget is reached.

Strategy 3: Going out strongThe mixed strategy puts either 50 000 or 100 000, and will place a new bid assoon as a new bid has entered or every second. If there is only one competitorleft, he will bid until the budget is reached.

Strategy 4: Reacting quicklyThis mixed strategy will make a bid as soon as another bid is placed. Hewill place bids between 5 000 - 30 000.

Strategy 5: Making irrational movesThis mixed strategy will place bids between 5 000-100 000.

Strategy 6: Start bidding lateThis pure strategy will wait until there is only one competitor left. He willthen place bids between 5 000-100 000.

4.2 The game

The game is made from the strategies that have been found and analyzed bythe real estate broker Asa Larsson [10]. The game is constructed in the waythat the players are rational, and stick to either a mixed or pure strategythroughout the game. Each strategy is assigned to one player. The strategiesthat have been used in the game are; putting everything on one card, layinglow, going out strong, reacting quickly, making ’irrational’ moves, and startbidding late. The strategies are mixed or pure depending on the outline ofthe strategy, though in the game representation, some of the pure strategiesare constructed like mixed strategies, as if the player has chosen to play soto make it more realistic.

The different strategies have individual budgets, which is also their real valueof the object. For each game, the players will get a budget between 400 000- 5 000 000 kr. The minimum amount of capital (budget) a player can get

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is 1.1 times the amount of the starting bid. The highest capital a playercan get is 1.4 times the amount of the starting bid. These values come fromthe analysis of bidding lists that we got from different real estate agenciesin Stockholm and Gothenburg. When the budgets have been randomly dis-tributed to the players, they will be randomly selected to start and who willplay next (make the next bid) to make it as close to reality as possible.

5 Results and analysis of the game

The game got constructed in the way that we have 6 different strategies andeach strategy represents one player. By these conditions we wanted to seehow they would behave against each others and which of them that wouldwin the most games. First we gave them randomly distributed budgets anda randomly chosen starting price. We chose to run the game 1 000 000 times.By figure 2 we can see the result that we got. We can see that none of thestrategies is better than the other. What we can see is that strategy 3 winsslightly less than the others and that strategy 1 wins slightly more. Theseresults are pretty obvious, since for all the games, the strategy that has beendistributed with the highest budget will win the game.

Figure 2: Randomly distributed budgets

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We also wanted to see how the strategies would behave if they got the samefixed budget. This time we also run the game 1 000 000 times. As we can seein figure 3, strategy 1 wins the majority of the games. But that is also notso surprising since strategy 1 is a very aggressive strategy and since they allhave the same budget and strategy 1 has the strategy to place his maximumbudget, it is natural that he will win and therefore a Nash equilibrium. Wecan also see that strategy 6 is never winning. That is because he has thestrategy to start bidding late, and since strategy 1 is so aggressive, he willnever have the chance to place a bid.

Figure 3: Fixed budgets

But how much is each strategy paying for an object in average? And which ofthem is paying less in average for an object than the others? When runningthe game randomly we can see (and understand from the previous resultsfrom the randomly played game) that they are paying in average about thesame.

Strategy 1 Strategy 2 Strategy 33 571 700 3 548 100 3 722 500

Strategy 4 Strategy 5 Strategy 63 569 900 3 540 800 3 564 700

Table 1: Average price paid from randomly played game

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If we take the average of the results in table 1, we can see more clearly thedifferences between the strategies. The average of the results is 3 051 616,66.What we can see is that the strategies is approximately paying the sameamount in average as each others. Which we also could see from figure 1.The only strategy that stands out is the strategy 3. Why he is paying morein average than the other strategies is because his strategy is to go out strongby putting high bids and continue to do so throughout the game, which inthe end will lead to paying much more for an object than needed.

To look at average prices for fixed budgets is unnecessary since it will alwaysshow the amount that we have chose the budget to be and is not providingany useful information.

To fully see and understand how a game can look like we have made an ex-ample. The example shows a game where the players have been distributedwith random budgets, and also randomly chosen to place their bids.

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Randomly given budget for each strategy:Strategy 1: 2 650 000 Strategy 2: 3 065 000 Strategy 3: 3 330 000Strategy 4: 3 115 000 Strategy 5: 3 130 000 Strategy 6: 2 995 000

Opening price: 2 405 000

Strategy(Player): Bid:4 2 430 0001 2 620 0004 2 640 0001 2 650 0004 2 675 0003 2 725 0002 2 735 0005 2 755 0003 2 855 0004 2 860 0005 2 900 0004 2 925 0003 2 975 0002 2 985 0005 3 065 0003 3 115 0005 3 125 0003 3 225 000

Table 2: Bidding list with random budgets

When looking at the budgets and table 2, we can see already in the beginningthat strategy 6 will never make a bid. His strategy of waiting until there isonly one competitor left does not hold, since his budget is lower than mostof his competitors he can not reach until the end of the game. Strategy 1will drop out after he reaches his budget and can not afford making any newbids. Strategy 2 drops out when strategy 5 bids his budget, and the winnerof this game is strategy 3, which also had the highest budget.

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6 Conclusion

The aim of this thesis was to see how players with different strategies wouldbehave when they were bidding on an object in the housing market, and alsoto see how they would behave if they play against a player with the samestrategy.

When we built the game we had the aim to have 10 different strategies and15 different players that could choose from these strategies, though duringthe time we encountered some obstacles. Some of the strategies could not beprogrammed randomly, but needed to get priority before the others, and thatwould not make the game random as we wanted. We also wanted the theplayers to meet opponents with the same strategy, but there were also thecase that they needed to get priority. So our final game became to include 6strategies which all became an individual player.

From our game we got the results that there are a difference between thefixed and the random played game. Our result showed that strategy 1 wasthe strategy that had the most winning auctions. In the fixed budget, strat-egy 1 had more than 60% of the winning auctions. But these results arepretty obvious. In the game with fixed budgets it is obvious that strategyhave the most wins since he will aways place his maximum budget which isthe other players budgets also, and they cannot place a higher bid, and hewill then win. In the game played with randomly distributed budgets it isalso obvious that there is not a clear winner who is the most efficient strategy,since it will always end with the players who have the highest budget.

From the paper Auction Theory: A Guide to the Literature by Paul Klem-perer [8], we can see that other researches also got the result that (as strategy1 has) the best strategy is to bid your true value of the object. He states thatthe most common winning strategy in ascending auctions is to bid the truevalue of the object, which here is representing the budget. Since strategy1 is very straight forward in the game with a pure strategy set of only twostrategies, bidding 30 000 less than budget, and bidding the budget value, itis clearly an aggressive strategy for winning when the budget is fixed. Sincethe rest of the strategies are not as aggressive with their strategies and bid-ding values, their chance of winning is less than for strategy 1.

Even though our results show us that the most common strategy to wina bidding game is by bidding the true value, it will be different in a realsetting with human interactions. In the real world there are factors that we

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could not include in our strategies. One factor that we could not includewas that bidders can change their strategies during the game after seeinghow the other bidders behave. In our game, the information is fixed fromthe start, which means that new information does not affect the game, theplayers will follow the game as if it is a script. Other factors include that thebidders can make bids before the bidding game has started, bidding on morethan one object, crossing borders and ask the real estate agent to sleep onit over the weekend to get more time. Another factor is that a bidder withhigher budget than other bidders, can choose to drop out anyway, moreover,the bidders in an auction are more likely to want to win below budget. Ifwe would have included these factors in our programming it would not havebeen a fair game, since we would be needed to give some strategies prioritybefore others. We wanted the game to be as fair as possible and chose thento exclude these types of strategies.

Limitations

Some limitations that we encountered during our research was that due tosecrecy we could not interview bidders from the bidding lists that we gotfrom the real estate agents. Since we could not interview the bidders, we didnot have any information about their budgets and strategies. The limitedinformation could not give us answers about why they dropped out from theauction. Maybe they reached their budget and if not, it would have beenuseful to know the reason they chose to leave the bidding anyway. From arational perspective, they could have chosen to leave due to lack of interestor a winning object in another auction, but it could also have been fromthe stressful mind games that happens in auctions that affect the biddersbehaviours. It would had been interesting to interview the bidders whattheir thoughts and strategies were during their bidding. For future researchin this subject, to really understand and perhaps find out what strategy isa truly winning strategy, is to interview people. To go to open houses andask what their budgets are and what strategy they would use to win theobject and follow the auctions with this insight information to be able tofully analyze from a game theoretic perspective.

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References

[1] Chatain, O. (2014) Cooperative and Non-cooperative Game Theory.London: Macmillan Publishers, 1-3

[2] Easley, D., Kleinberg, J. (2010) Networks, Crowds, and Markets:Reasoning about a Highly Connected World. Cambridge, UK: CambridgeUniversity Press, 209-210

[3] Geckil, I K., Anderson, P. L. (2010) Applied game theory and strategicbehavior. Boca Raton, FL: Chapman and Hall/CRC, 3, 9, 12-14, 16-19,25-26

[4] Gintis, H. (2000) Classical versus evolutionary game theory. Journal ofConsciousness Studies Vol. 7 (1-2) Exeter, UK: Imprint Academic

[5] Hansson, T. (2016) How the Swedish housing market works, perspectivesfrom Thomas Hansson, real estate agent at Maklarhuset, Askim. Inter-viewed by Sofia Hjulfors and Malin Norling, April 2016. Askim

[6] Hargreaves Heap, S. P., Varoufakis, Y. (2004) Game theory: A Criticaltext. London: Psychology press, 44, 59

[7] Holzman, R. (2002) Foundations of non-cooperative games. Oxford,UK: Encyclopedia of Life Support Systems (EOLSS), UNESCO/EolssPublishers.

[8] Klemperer, P. (1999) Auction Theory: A Guide to the Literature.Journal of Economic Surveys, Vol. 13, Issue 3. Oxford, UK: Blackwellpublishers Ltd, 227-286

[9] Krishna, V. (2002) Auction Theory. New York, USA: Elsevier Science,279-284

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[10] Larsson, A. (2013) Budgivningsstrategier. Maklarvardag [Blog] Availableat: http://maklarvardag.blogspot.se/2013/05/budgivningsstrategi.html[Accessed 3 Apr. 2016]

[11] McKenzie, A. J. (2009) Evolutionary Game Theory. TheStanford Encyclopedia of Philosophy. [online] Stanford, USA:Metaphysics Research Lab, Stanford University. Available at:https://plato.stanford.edu/archives/fall2009/entries/game-evolutionary[Accessed 2 Feb. 2017]

[12] Bergemann, D. (2006) Game Theory and Information Economics, De-partment of economics, Yale Univrsity

[13] Osborne, M. J., Rubinstein, A. (1994) A course in game theory.Cambridge, USA: The MIT Press

[14] Prisner, E. (2014) Game Theory: Through examples. Washington, D.C.:The Mathematical Association of America, 1-2

[15] Rasmusen, E. (2001) Games and information: An Introduction toGame Theory. Third edition. Cambridge, MA: Blackwell Publisher, 19,31-35, 177

[16] Samsura, D.A.A., Krabben, E. van der and Deemen, A.M.A. van(2010) A game theory approach to the analysis of land and propertydevelopment processes. Land Use Policy, Vol 27, Issue 2. Oxford, UK:Elsevier BV, 564-578.

[17] Shoham, Y., Leyton-Brown, K. (2009) Multiagent Systems. Cambridge,England: Cambridge University Press, 56, 62, 118-119, 130-137, 143-144,163-168, 224-225, 228, 329

[18] Turocy, T., von Stengel, B. (2002) Game Theory. Encyclopia of Infor-mation Systems, Volume 2. San Diego: Elsevier Science, 403-420

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A Appendix: MATLAB codes

A.1 The auction

1 c l c2 c l e a r a l l3 rounds =1;4 win=[0 0 0 0 0 0 ] ;5 Average win =[0 0 0 0 0 0 ] ;6 name={ ’ S t rategy 1 ’ , ’ St rategy 2 ’ , ’ St rategy 3 ’ , ’

St rategy 4 ’ , ’ St rategy 5 ’ , ’ St rategy 6 ’ } ;7

8 whi le rounds ˜= 1000019 % % % % % % % % % % % % % % % % % %

10 % s t a r t i n g b i d = 500 ;11 % s c a p i t a l = [1000 1000 1000 1000 1000 1 0 0 0 ] ;12 % % % % % % % % % % % % % % % % % %13 s t a r t i n g b i d = round ( randi ( [ 4 00 5000 ] , 1 , 1 ) /5) ∗5 ;14 % The lowest amount o f c a p i t a l a person can have :15 min cap i t a l = round (1 . 1∗ s t a r t i n g b i d ) ;16 % The h ighe s t amount o f c a p i t a l a person can have :17 max capi ta l = round (1 . 4∗ s t a r t i n g b i d ) ;18 % randomly d i s t r i b u t e s c a p i t a l to each bidder19 % between min and max o f a v i a b l e c a p i t a l20 s c a p i t a l = round ( randi ( [ m in cap i t a l max capi ta l ] , 1 , 6 )

/5) ∗5 ;21

22 % % % % % % % % % % % % % % % % % %23 b i d s a r r a y= ze ro s (1 , 7 ) ;24 compet i tor s = 6 ;25 n =0;26

27 o l d b i d=s t a r t i n g b i d ;28 auc t i ons =1;29

30

31

32 whi le compet i tor s ˜= 133 compet i tor s=c o m p e t i t o r s l e f t ( s c a p i t a l ) ;34 switch n35 case 1

28

36 i f ( b i d s a r r a y (n) ˜= 0 && s c a p i t a l (n) ˜= 0) | |b i d s a r r a y (7 )==0

37 [ c u r r e n t b i d , s c a p i t a l (n) , b i d s a r r a y ] = s1 (n, cur r ent b id , s c a p i t a l (n) , b id s a r ray ,compet i tor s ) ;

38 e l s e39 end40

41 case 242 i f ( b i d s a r r a y (n)˜= 0 && s c a p i t a l (n) ˜= 0 &&

c u r r e n t b i d ˜= s t a r t i n g b i d )43 [ c u r r e n t b i d , s c a p i t a l (n) , b i d s a r r a y ] = s2 (n

, cur r ent b id , s c a p i t a l (n) , b id s a r ray ,compet i tor s ) ;

44

45 e l s e46 end47 case 348 i f ( b i d s a r r a y (n) ˜= 0 && s c a p i t a l (n) ˜= 0) | |

b i d s a r r a y (7 )==049 [ c u r r e n t b i d , s c a p i t a l (n) , b i d s a r r a y ] = s3 (n


50




56




29

62

63 e l s e64 end65 case 666 i f b i d s a r r a y (n) ˜= 0 && s c a p i t a l (n) ˜= 067 [ c u r r e n t b i d , s c a p i t a l (n) , b i d s a r r a y ] = s6 (n


68

69 e l s e70 end71 otherw i s e72 c u r r e n t b i d=s t a r t i n g b i d ;73 end74

75 % % % % % % % % % % % % % % % % % %76 % Auction l i s t77 i f c u r r e n t b i d˜=o l d b i d && rounds==10000078 b i d l i s t ( auct ions , 1 )=n ;79 b i d l i s t ( auct ions , 2 )=c u r r e n t b i d ;80 o l d b i d=c u r r e n t b i d ;81 % checking the s t a r t i n g c a p i t a l on the round82 i f auc t i ons==183 Au ct i on s t a r t i ng b id=s t a r t i n g b i d ;84 s t a r t i n g c a p i t a l=repmat ( s c a p i t a l , 1 ) ;85 end86 auc t i ons=auct i ons +1;87 end88 % % % % % % % % % % % % % % % % % %89 n = randi (6 ) ;90

91 end92 [ win]= winner ( s c a p i t a l , win ) ;93 [ Average win ] = Average ( s c a p i t a l , Average win ) ;94 rounds=rounds +1;95 end96 Average win=Average win . / win ;97

98 p i e ( win )99 l egend (name( win>0) )

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A.2 Bid counter

1 f unc t i on [ b i d s a r r a y ] = b id counte r ( n , b i d s a r r a y )2 % This func t i on i s a bid counter where the f i r s t3 % s i x d i g i t s o f the array i s count ing the4 % i n d i v i d u a l b ids done by othe r s s i n c e the l a s t5 % time the compet i tor did a bid . The seventh d i g i t6 % i s the t o t a l sum of b ids done in the game .7

8 b i d s a r r a y (1 )= b i d s a r r a y (1 ) +1;9 b i d s a r r a y (2 )= b i d s a r r a y (2 ) +1;

10 b i d s a r r a y (3 )= b i d s a r r a y (3 ) +1;11 b i d s a r r a y (4 )= b i d s a r r a y (4 ) +1;12 b i d s a r r a y (5 )= b i d s a r r a y (5 ) +1;13 b i d s a r r a y (6 )= b i d s a r r a y (6 ) +1;14 b i d s a r r a y (7 )= b i d s a r r a y (7 ) +1;15

16 switch n17 case 118 b i d s a r r a y (1 ) =0;19 case 220 b i d s a r r a y (2 ) =0;21 case 322 b i d s a r r a y (3 ) =0;23 case 424 b i d s a r r a y (4 ) =0;25 case 526 b i d s a r r a y (5 ) =0;27 case 628 b i d s a r r a y (6 ) =0;29 case 730 b i d s a r r a y (7 ) =0;31 end

31

A.3 Competitors left

1 f unc t i on [ compet i tor s ] = c o m p e t i t o r s l e f t ( s c a p i t a l )2 compet i to r s out = 0 ;3 f o r i = 1 : l ength ( s c a p i t a l )4 i f 0==s c a p i t a l ( i )5 compet i to r s out = compet i to r s out +1;6 end7 end8 compet i tor s=6−compet i to r s out ;9 end

A.4 Winner

1 f unc t i on [ win ] = winner ( s c a p i t a l , win )2

3 win=win+( s c a p i t a l >1) ;4

5 end

A.5 Average

1 f unc t i on [ Average win ] = Average ( s c a p i t a l , Average win ,c u r r e n t b i d )

2 i =1;3 whi le s c a p i t a l ( i )==04

5

6 i=i +1;7 end8 s c a p i t a l ( i )=c u r r e n t b i d ;9 Average win=Average win+s c a p i t a l ;

10

11 end

32

A.6 Strategy 1

1 f unc t i on [ output bid , s c a p i t a l , b i d s a r r a y ] = s1 ( n ,cur r ent b id , s c a p i t a l , b id s a r ray , compet i tor s )

2 % Strategy 1 : Putt ing everyth ing on one card3 % ( Pure s t r a t e g y )4 % Put max( budget )−30 000 . 30 000 (30 K)5 % i s based on p o s s i b i l i t y to play one more round6 % i f other p l aye r s respond to the new auct ion p r i c e .7 % Response : I f next bid i s lower than 30 K from the

prev ious ,8 % bid again up to the budget p r i c e .9

10 i f c u r r e n t b i d >= ( s c a p i t a l − 30) && c u r r e n t b i d <s c a p i t a l ;

11 output b id = s c a p i t a l ;12 b i d s a r r a y = bid counte r (n , b i d s a r r a y ) ;13 e l s e i f c u r r e n t b i d +30 < s c a p i t a l ;14 output b id = s c a p i t a l − 30 ;15 b i d s a r r a y = bid counte r (n , b i d s a r r a y ) ;16 e l s e i f c u r r e n t b i d >= s c a p i t a l17 s c a p i t a l = 0 ;18 output b id = c u r r e n t b i d ;19 e l s e20 output b id=c u r r e n t b i d ;21

22

23 end24 end

33

A.7 Strategy 2


2 % Strategy 2 : Laying low ( Pure s t r a t e g y )3 % After the f i r s t rounds , s t a r t b idding 5 K−10 K4 % to make the bidding go s lower and making people5 % nervous to get to the end .6 % Maybe other p l aye r s w i l l win another auct ion whi le7 % s t i l l b idding on the one happening now .8 % Response : Wait f o r 2 p l aye r s to bid be f o r e9 % making a new bid (10 K) . I f the re i s only one p laye r

10 % l e f t , bid every time a new auct ion bid has been made. ( 5 K)

11

12

13 a v a i a b l e b i d s =[5 1 0 ] ;14

15

16 i f b i d s a r r a y (2 )>=2 && s c a p i t a l>= c u r r e n t b i d+a v a i a b l e b i d s (2 ) ;

17 output b id = c u r r e n t b i d+a v a i a b l e b i d s (2 ) ;18 b i d s a r r a y = bid counte r (n , b i d s a r r a y ) ;19

20 e l s e i f compet i tor s==2 && s c a p i t a l>= c u r r e n t b i d+a v a i a b l e b i d s (1 ) ;

21 output b id = c u r r e n t b i d+a v a i a b l e b i d s (1 ) ;22 b i d s a r r a y = bid counte r (n , b i d s a r r a y ) ;23

24 e l s e i f c u r r e n t b i d >= s c a p i t a l−a v a i a b l e b i d s (1 )25 s c a p i t a l = 0 ;26 output b id = c u r r e n t b i d ;27 e l s e28 output b id=c u r r e n t b i d ;29

30 end31

32 end

34

A.8 Strategy 3


2 % Strategy 3 : Going out s t rong ( Mixed s t r a t e g y )3 % Making high b ids though out the auct ion ,4 % making people scared o f en t e r i ng ? 50 K−100 K.5 % Response : Make a p r o b a b i l i t y d i s t r i b u t i o n 1/26 % f o r p lay ing every time a new bid has a r r i v e d7 % and every second new bid has a r r i v ed .8 % I f the re i s only one p laye r l e f t ,9 % bid every time a new auct ion bid has been made .

10

11 a v i a b l e b i d s =[50 1 0 0 ] ;12

13

14 i f c u r r e n t b i d < s c a p i t a l && s c a p i t a l >= c u r r e n t b i d+a v i a b l e b i d s (1 ) && compet itors >=2;

15 i f b i d s a r r a y (3 )==116 bid= randi (2 )−1;17 i f bid==1 && s c a p i t a l >= c u r r e n t b i d+

a v i a b l e b i d s (2 )18

19 output b id = c u r r e n t b i d+a v i a b l e b i d s ( randi(2 ) ) ;

20 b i d s a r r a y = bid counte r (n , b i d s a r r a y ) ;21 e l s e i f bid==1 && s c a p i t a l >= c u r r e n t b i d+

a v i a b l e b i d s (1 )22 output b id = c u r r e n t b i d+a v i a b l e b i d s (1 ) ;23 b i d s a r r a y = bid counte r (n , b i d s a r r a y ) ;24

25 e l s e26

27 output b id=c u r r e n t b i d ;28

29 end30 e l s e31 output b id = c u r r e n t b i d+a v i a b l e b i d s ( randi (2 ) ) ;32 b i d s a r r a y = bid counte r (n , b i d s a r r a y ) ;33 end34

35

35

36 e l s e i f s c a p i t a l >= c u r r e n t b i d+a v i a b l e b i d s (2 ) &&compet i tor s==2;

37 output b id = c u r r e n t b i d+a v i a b l e b i d s ( randi (2 ) ) ;38 b i d s a r r a y = bid counte r (n , b i d s a r r a y ) ;39 e l s e i f s c a p i t a l >= c u r r e n t b i d+a v i a b l e b i d s (1 ) &&

compet i tor s==2;40 output b id = c u r r e n t b i d+a v i a b l e b i d s (1 ) ;41 b i d s a r r a y = bid counte r (n , b i d s a r r a y ) ;42 e l s e i f c u r r e n t b i d >= s c a p i t a l−a v i a b l e b i d s (1 )43 s c a p i t a l = 0 ;44 output b id = c u r r e n t b i d ;45 e l s e46 output b id=c u r r e n t b i d ;47

48

49 end50

51

52 end

36

A.9 Strategy 4


2 % Strategy 4 : Reacting qu i ck ly ( Mixed s t r a t e g y )3 % Making a new bid as soon as another bid e n t e r s4 % with a p r o b a b i l i t y d i s t r i b u t i o n over 5K − 30 K.5 i =6;6 whi le s c a p i t a l <= c u r r e n t b i d+i ∗57 i f s c a p i t a l >= c u r r e n t b i d+i ∗58 break9 e l s e i f i==1

10 break11 e l s e12 i= i −1;13 end14 end15 p l a c i n g b i d=c u r r e n t b i d+randi ( i ) ∗5 ;16

17 i f c u r r e n t b i d >= s c a p i t a l −518 s c a p i t a l = 0 ;19 output b id = c u r r e n t b i d ;20 e l s e i f c u r r e n t b i d < s c a p i t a l && s c a p i t a l >=

c u r r e n t b i d+i ∗521 output b id = c u r r e n t b i d+randi ( i ) ∗5 ;22 b i d s a r r a y = bid counte r (n , b i d s a r r a y ) ;23

24 e l s e25 output b id=c u r r e n t b i d ;26

27 end

37

A.10 Strategy 5


2 % Stratey 6 : Making i r r a t i o n a l moves3 % ( Mixed s t r a t e g y )4 % Making uneven b ids with a p r o b a b i l i t y5 % d i s t r i b u t i o n over a s e t o f numerica l6 % values to make changes in the auct ion7 % and make other p l aye r s confused8 % with t h e i r own bidding s t r a t e g i e s .9 % 5 K−100 K depending on budget and time o f game .

10

11 i =20;12 whi le s c a p i t a l <= c u r r e n t b i d+i ∗513 i f s c a p i t a l >= c u r r e n t b i d+i ∗514 break15 e l s e i f i==116 break17 e l s e18 i= i −1;19 end20 end21

22 p l a c i n g b i d=c u r r e n t b i d+randi ( i ) ∗5 ;23

24 i f c u r r e n t b i d >= s c a p i t a l −525 s c a p i t a l = 0 ;26 output b id = c u r r e n t b i d ;27 e l s e i f c u r r e n t b i d < s c a p i t a l && s c a p i t a l >=

c u r r e n t b i d+i ∗528 output b id = c u r r e n t b i d+randi ( i ) ∗5 ;29 b i d s a r r a y = bid counte r (n , b i d s a r r a y ) ;30

31 e l s e32 output b id=c u r r e n t b i d ;33 end

38

A.11 Strategy 6


2 % Strategy 7 : S ta r t b idding l a t e ( Mixed s t r a t e g y )3 % Waiting u n t i l the r e i s only two p laye r l e f t in4 %the auct ion i f the p r i c e i s s t i l l below the5 % budget . P rob ab i l i t y d i s t r i b u t i o n over 5 K−100 K.6 i =20;7 whi le s c a p i t a l <= c u r r e n t b i d+i ∗58 i f s c a p i t a l >= c u r r e n t b i d+i ∗59 break

10 e l s e i f i==111 break12 e l s e13 i= i −1;14 end15 end16 p l a c i n g b i d=c u r r e n t b i d+randi ( i ) ∗5 ;17

18 i f c u r r e n t b i d >= s c a p i t a l −519 s c a p i t a l = 0 ;20 output b id = c u r r e n t b i d ;21 e l s e i f p l a c i n g b i d <= s c a p i t a l && compet i tor s <= 3 ;22 output b id = p l a c i n g b i d ;23 b i d s a r r a y = bid counte r (n , b i d s a r r a y ) ;24

25 e l s e26 output b id=c u r r e n t b i d ;27

28 end

39

Bidding strategies in the Swedish housing marketumu.diva-portal.org/smash/get/diva2:1093030/FULLTEXT01.pdf · game will therefore depend on the players’ choices of strategic or

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