Bidding Strategies in the Swedish Housing Market

Bidding strategies in the Swedish housing market

Malin Norling Soﬁa Hjulfors

Autumn semester 2016 Bachelor thesis, 15 hp Bachelor degree of mathematics, 180 hp Department of mathematics and mathematical statistics Acknowledgements

We both would like to thank the real estate agencies and the people who helped us and provided bidding lists that we could analyze; Fastighetsbyr˚an- Mölnlycke, Fastighetsbyr˚an- Bromma, Fastighetsbyr˚an- Kungsängen, Mäklarhuset- Askim. We would specially like to thank Thomas Hansson from Mäklarhuset, Askim, who we got the opportunity to interview to get a bigger understanding of the 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 was and still is my role model. I also would like to thank my mom Anett, my sister 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 friends and family. I would specially like to thank my father, Stefan, for pushing us both during this period, to inspire us to think outside the box. I would like to thank Pinja, may she rest in peace, that took us on inspiring walks when we needed breaks. - Sofia Hjulfors Abstract This report focuses on an introduction of game theoretical models and how they can be applied in the Swedish housing market. Game theory is a study of mathematical models of human conflicts and cooperation between rational decision makers within a competitive situation. There are several different strategies that a player can use. In this thesis each strategy is assigned to one player. So how will the players behave in a game, and what strategy is the most successful? By using the software MatLab, the authors creates a game where the strategies assigned to each player gets randomly distributed budgets and are randomly selected to place bids during the game. The game is then played 1 000 000 times to see what strategy is the most successful. It is also tested to see what strategy is the most successful if the players have the same budgets. The authors conclude that in practice it is the size of the budget that determines who will win the bidding, hence there are minor differences between the different strategies in how much they pay on average to win.

Sammanfattning Denna rapport fokuserar p˚aen introduktion av spelteoretiska modeller och hur de kan som kopplas till den svenska bostadsmarknaden. Spelteori är en studie om matematiska modeller för mänskliga konflik- ter och samarbete mellan rationella beslutstagare i en konkurrensut- satt marknad. Det finns flera olika strategier en spelare kan använda sig av. I denna rapport blir varje spelare tilldelad en strategi. S˚afr˚agan ställs hur spelarna kommer att bete sig och vilken av strategierna som är den mest framg˚angsrika. Genom att använda programvaran MatLab, skapar författarna ett program där varje strategi är tilldelad varje spelare och som helt slumpmässigt f˚aren budget och även blir slumpmässigt valda att spela, d.v.s lägga bud under spelets g˚ang.Spe- let spelas därefter 1 000 000 g˚angerför att se vilken av strategierna som är mest framg˚angsrik.Det är även testat att se vilken strategi som f˚arbäst resultat om de alla har samma budget. Författarna drar slutsatsen att i praktiken är det storleken p˚abudgeten som bestämmer vem som vinner budgivningen, dock att det även finns mindre skillna- der mellan strategierna som bestämmer hur mycket de i genomsnitt f˚arbetala när 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 3 2.1 Rules of the game ...... 3 2.1.1 Cooperative and non-cooperative games ...... 5 2.1.2 Simultaneous and sequential games ...... 6 2.2 Normal form games ...... 7 2.3 Nash equilibrium ...... 7 2.4 Extensive form games ...... 8 2.4.1 Extensive games with perfect information ...... 8 2.4.2 Extensive games with imperfect information ...... 10 2.5 Bayesian games ...... 10 2.6 Evolutionary game theory ...... 12 2.7 Auction theory ...... 13

3 The housing market 14 3.1 Interview with a real estate agent ...... 14 3.2 Auction rules ...... 15 3.3 Bidding strategies ...... 16

4 The auction game 18 4.1 Rules of the game ...... 18 4.2 The game ...... 19

5 Results and analysis of the game 20

6 Conclusion 24

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

1 The Sharing game ...... 9 2 Randomly distributed budgets ...... 20 3 Fixed budgets ...... 21

List of Tables

1 Average price paid from randomly played game ...... 21 2 Bidding list with random budgets ...... 23 Nomenclature

∈ Element of ∪ Union ∩ Intersection ∅ Empty set × Cartesian product R Real numbers G Game i Player n Number of players N Set of all players, N = 1, ..., n Si Set of pure strategies for each player i ∈ N,Si = s1, ..., sm ∗ ∗ ∗ ∗ Si Set of mixed strategies for each player i ∈ N,Si = s1, ..., sn 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 nodes Z Set of terminal nodes, Z ∩ H = ∅ P Probability vector, P = (p1, ..., pm) U Payoﬀ (utility) α Action function ρ Player function σ Successor function 1 Introduction

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

Auctions are a particular type of game that are studied in a branch of game theory, which is known as auction theory. In many real-life situations we can find auctions and they are often found when dealing with economic transactions. Auction theory is therefore an important tool for understanding interaction among sellers and buyers. There are many different sets of rules for an auction that defines what type of auction it is. There is for example the sealed-bid auction of first and second price, the descending-price auction, and the Japanese auction [17]. In this thesis, the authors will focus on the auction model for open ascending bids, which is the model used in the Swedish housing market.

A potential value of applying auction theory to the housing market is to give real estate agents a better understanding of how the market work and also the possibilities to analyze consequences of possible rule changes. It is also important for the buyers in the housing market to understand how to get an 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 rules can 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 auction theory applies. How should a game-theoretical model for the Swedish

1 housing market be constructed?

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

The problem formulation for this thesis has been to see if there is a game theoretical 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 mathematical models of human conflict and cooperation between rational decision makers within a competitive situation. When implementing it into the Swedish housing market and the bidding game, the authors used the game theory methods; extensive form, perfect and incomplete information, mixed and pure strategies, non-cooperative, and n-player. The authors chose these methods to show the connection related to the Swedish housing market, furthermore an introduction of evolutionary game theory and auction theory, which are all explained in section 2. When constructing the game, the authors needed to find all different strategies that bidders can use in auctions in the Swedish housing 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 explains the author’s game that is implemented in MatLab.

2 2 Game theory

This chapter will give a brief introduction to some of game theory’s concepts that 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 elements consist of players, information, actions and payoﬀ which all depends on the particular game that is played, that usually is of normal form or extensive form. This general description of the elements and rules will be used for eval- uating our own game-theoretical model in the housing market [16], including some brief introduction of other types of game theoretical models.

Players The players, denoted N, consist of the number of people that are playing the 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 to maximize its utility. Each player knows that the other players in the game also have the goal to maximize its utility [15], though the utility might be of diﬀerent value among the players. Moreover, the players’ interests could both be matching or conﬂicting, meaning that the players can cooperate or not to reach utility.

Information The rules of the game decide what kind of information about diﬀerent variables is available to the players [3]. The information, denoted I, can also change during the game. Games where perfect information occur give all the players the same information throughout the game [3]. The players have knowledge about the other players moves like in a game of chess where all the chess pieces are visual for both players throughout the game. In contrary, imperfect information has the opposite eﬀect. The players cannot see the other players moves.

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

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

How well the players know the structure depends on if the game has complete or incomplete information [14]. Given a game with complete information, the players are aware of the structure of the game. The players are aware of how the game is played, all possible actions and the payoffs for those outcomes. A game with incomplete information is more complex since there are many variables unknown. The players might be unaware of how many players there are in the game, what their payoffs are and what actions are possible. They are 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 other players.

Actions The actions, denoted A, are the moves that the players will do depending on previous actions from other players, information available, preferences, rules etc. Each game has a number of rules to follow to be able to make actions in the game. Rules might include perfect or imperfect information, availability or 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 when the game has finished [3]. This depends on the strategies chosen during the game, the actions made by other players and other variables tied to the specific game. Predicting the payoff in a real life situation is almost impossible due to the possibility of irrational players. In a game-theoretical model it is possible to predict the players’ possible outcomes and expected utility when it exists perfect and complete information due to the shared information between the players.

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

4 game with pure strategy the player decides a strategy profile to use throughout the game. This profile has rules for every possible action made of others in the game, making the action process fixed for every node in the game. In a game with mixed strategy the player choose a probability distribution over the set of pure strategies and the strategies become randomized rather than fixed [18].

Definition 1. If a player, i has a set of Si pure strategies {s1,...,sm}, a mixed ∗ strategy set, Si is defined by the probability vector, P = (p1, ..., pm), where ∗ the players’ strategy will be selected. Observe that for Si 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 mixed strategy (p1 = 0, ..., p2 = 1, ..., pm = 0), it is equivalent to choosing the pure strategy set S2 [6].

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

Cooperative game theory The cooperative game theory is a model that states what payoffs each group of players, or coalition, obtains in the cooperation of the players [1]. It is not definitive that the process forms the coalition, which can be when the players are in different groups in an parliament [18]. Every player in the group have different 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 process a 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 players are involved in an interactive process whose outcome is determined by each players’ decision [7]. The players are not allowed to make binding agreements with other players. They can communicate with each other and discuss

5 united strategies of actions to the game, but during the game the players are not allowed to communicate to each other and are then autonomous decision makers [7]. The theory consists of three main subjects. The ﬁrst one is the progress of technical models that build frame works for the representation of the game. The second is the concept that engages the idea of rational behaviour, and the third and last is the mathematical tools that is used in order to demonstrate statements of the concepts of the existence and the equilibrium [7].

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

Sequential game theory In the sequential game theory the players take interchanging turns to make their choices. In sequential games it is important to know who is going to make the ﬁrst move [3], since it can be both an advantage and an disadvantage to start. Depending on the game structure and information, the players might know the actions taken in the previous turns and be able to observe each other and follow the other players future moves. If this is true, the player can use the information to estimate their own upcoming move, and this is called the “backward induction”. An example for this is the game chess. When making the move, the player knows the previous moves of the other player and can use that to predict the next move. In that way, the player 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 same time [3]. As long as they do not know about the other players’ choices in action, they do not need to make their decisions at the same time. It can be very diﬃcult in this type of game since the players do not know what the other players strategies are.

6 2.2 Normal form games The normal form game, also called the strategic form game, is a game G with a ﬁnite set of N players [13]. The game is played simultaneous at a one-time decision 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 of action proﬁles.

Deﬁnition 2. A normal form game is a tuple hN, A, ui 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 actions available 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 concept of the Nash equilibrium of a strategic game. It assumes that each player’s strategy maximizes his payoff [3]. The Nash equilibrium, also called the strategic equilibrium, captures therefore a steady state of the play of the game [13]. All players expect that the other players will follow the strategy that maximizes his payoff and that they cannot change their strategy unilaterally and get a better payoff. By separating the strategy of player i from all other players, it will be denoted by s−i. Definition 3. Strategy profile s∗ is a Nash equilibrium if, for all i = 1, ..., N ∗ ∗ ∗ and all si ∈ Si, ui(si , 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 the other players’ strategies. The weak Nash equilibrium is less stable than the strict, since at least one player has a best response to the other players’ strategies,which is not his equilibrium strategy. Depending on the game of pure strategy the Nash equilibrium can be either weak or strict, and in the game of mixed strategy the Nash equilibrium is weak [17].

7 2.4 Extensive form games Games in extensive form, introduced by John Von Neumann involves different stages called nodes, that can be presented in a tree graph [18]. At every node h the player can choose a strategy with knowledge from some information about the previous moves by other players. For this reason, the nodes can be 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. The structure of the game is the information, what each player knows about the other players and their moves, and the utility for all possible outcomes. Every game starts at the initial node and continues through the tree. Every non- terminal node belongs to one player. At that node, the player can choose between a number of different strategies, and each strategy leads to another node [18]. For games with perfect information, each node in the tree graph will represent all possible strategies that can be chosen by a player. The player can have a mixed or pure strategy profile, where the mixed strategy profile is represented by a probability distribution over a pure strategy set. The players in the game have complete knowledge about the history of actions of all players in the game and also at what node they are in [18]. Games with imperfect information have restricted information or no information concerning the previous actions and behaviour in the game and because of that, the players must make assumptions. The choice nodes are partitioned into information sets. With that being defined, the player cannot distinguish between 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 tuple G = (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 set of possible actions • A player function ρ : H 7→ N which assigns to each non-terminal node a player i ∈ N who chooses an action at that node

8 • A successor function σ : H × A 7→ H ∪ Z which maps a choice node and an action to a new choice node or terminal node such that for all h1, 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 Z u = (u1, ..., un) where ui : Z 7→ R

Figure 1: The Sharing game

Figure 1, the sharing game from previous page is a perfect-information game in extensive form with pure strategies [17]. The game shows the pure strategies available when two siblings, one brother and one sister, follow a protocol of 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 she accepts the split or not depending on the preferable outcome. If the sister would decide to not accept the brother’s split, they would both go empty handed. Looking at Figure 1, for the brother and sister to get gifts and be equally happy, the boy should suggest a split where they keep one gift each and the sister should accept the split. The Sharing game is furthermore a representation of three pure strategies for player 1, the brother, and eight pure 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)}

9 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 classes on {h ∈ H : ρ(h) = i} with the property that α(h) = α(h0) and 0 0 ρ(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 the choice nodes, the set of actions at each choice node in the information set must be the same. If Ii,j ∈ Ii is an equivalence class, then we can use the notation α(Ii,j) to describe all the actions available to player i at any node in the information set Ii,j [17].

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

2.5 Bayesian games Bayesian games introduce incomplete information in the game, where the structure of the game is not fully known by the players. Preferences, strategies, budgets and utility functions might be some of the information that is not shared among the players, making it more difficult to predict how the other players will act in the game and how to play against them [9]. Games with incomplete information were developed by Harsanyi (1967-1968) [9] who stated 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 know their own private type and not the others. Based on the players’ type they will make beliefs and assumptions about other players’ type. They will also make assumptions about how other players will see and respond to the individual player, their private utility function, the one of others and other assumptions related to the game [9]. The Bayesian game can be defined in different ways depending on the underlying structure. Besides looking at different type of players, Harsanyi found a way to reconstruct the concept

10 of incomplete information into complete and imperfect information, without changing the essential concept [9]. The information that is not shared among the players would be seen as though players have different information, i.e. imperfect information and that Nature plays a roll in the game leading to a hypothetical random first move [17]. Nature can be seen as a player with a mixed strategy profile to easier understand the concept. From here it is possible to analyze the game and find a Bayesian-Nash equilibrium which intuitively states that each player is doing his best to maximize his payoff given his beliefs and assumptions about the behavior of other players [17]. Bayesian games are found in many real-life and economic situations where the information about utility functions is partly or not at all shared, such as in non-cooperative house auctions for example. In that situation, the bidders in the auction are solely aware about their own preferences and the utility function regarding a certain house that they are bidding on, but not aware about the other bidders’ preferences and utility functions that are bidding for the same house. This makes in difficult for the bidders to make predictions of 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 the auction 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] that has:

• A set N of n players, indexed by i

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

11 2.6 Evolutionary game theory Evolutionary game theory apply game theoretical ideas to model human and animal behavior of strategic interaction. Some differences that divide evolutionary game theory from classical game theory is that the concept of rationality is more relaxed, yet providing valid conclusions. It focuses more on the dynamics of a population and natural selection, how individuals change over time through a replicator rather than the individuals payoff maximiza- tion of a game and equilibrium [4]. The focal point of rationality has also been proven by science in numerous testings that human actors are more often irrational than not, making evolutionary game theory more applicable to study human behaviors and interactions than game theory [11]. The theory was introduced and developed from evolutionary biology in 1930 [11] from which they originally looked at how different species and their genes determined characteristics of behaviours and how they evolve over time. These behaviours got translated into strategic behaviors and the decisions made where seen as unconscious decisions, mutations or selections and therefore not rational. Payoffs would be given when interactions among species oc- curred in a population and can therefore be seen as a game, where the fittest species would get the highest payoff by having a higher reproduction rate and drive out other species [2]. The strategic behaviors could be seen as a best response from a game theoretical perspective, but biologists were more interested 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 understanding and 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 a model where individuals frequently interact in an population which give them payoffs in terms of fitness. The main point of the replicator dynamics states that individuals make approximately copies of themselves by reproduction and that individuals with higher payoffs also reproduce faster than other individuals, making those individuals with a certain replicator-fitness to increase in the population. These replications can for example be a strategy in a game, cultural or a gene. Replications can also include random errors as mutations [17]. The static approach is related to the replicator dynamics and is called the evolutionarily stable strategy (ESS) which was introduced by Maynard Smith and Price [17]. Its stability concept states that if the majority of a population is using a certain mixed strategy, a small number of entering individuals using another mutant strategy cannot successfully in-

12 vade the population. The original strategy is therefore an evolutionary stable strategy, which results in a higher payoﬀ than the new mutant strategy, and will therefore drive out the new individuals from the population. This is strongly related to the Nash equilibrium in game theory. For larger numbers of individuals using a mutant strategy, they could be able to successfully invade the population [17].

2.7 Auction theory Auctions are widely used for diﬀerent economic transactions, computational settings, in war games and other real life situations. Auction theory has been an important tool for understanding interactions between sellers and buyers and moreover for understanding games of incomplete information in game theory [8]. There are many kinds of auction-theoretical models for understanding resource allocation problems between self-interested agents. The sealed-bid auction of ﬁrst and second price, the descending-price auction and the Japanese auction are only some of the standard models used in auction theory. Introducing the auction type of open ascending bids, we will focus on the auction model concerning the Swedish housing market in this research [17].

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

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

13 independent private values can be seen as games of incomplete information in 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 realtors Mäklarhuset in Askim, Gothenburg and interviewed the real estate agent and partner of Mäklarhuset, Thomas Hansson [5]. He informed us that the average of biddings on objects depends on the market conditions and what kind of objects are for sale. He also stated that if it is an attractive object there can be up to 45-50 families that are interested but it also depends on the value of the first bid. If the first bid is very high many speculators will drop out, hence if it is low or on the asking price more speculators will join the auction. The time period of the auction is normally 1-2 days. When the real 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 the area today etc. Thomas Hanson made it clear that the objects’ value is their correct one when they get listed on the market, that it is not a lower price to attract more speculators. He also pointed out that during periods when there are fewer object out on the market, the biddings tend to increase more than 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 changes in interest rates. The only time a seller is potentially making a faster sell than 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 before the scheduled viewing of the object. The real estate agents are trying to prevent it, but if the seller ﬁnds the early bid good he will probably accept even if no speculator has been there to see the property. Why this has become more common is probably that the buyer has earlier experienced this form 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 make early bids. To place a bid before the viewing is one type of strategy. But is there a strategy that is better than any other strategy? For all that Thomas knows, there is no strategy that is better than another. It all depends on the other buyers and the market at the speciﬁc time.

14 The limiting knowledge for speculators about other speculators can be con- fusing for the involved. During the auction process the speculators might think that they are only four speculators if only four speculators have entered the auction with bids. Sometimes in the end of the auction one or more speculators join. For the other speculators it seems as if they are new in the auction but the real estate agent has had knowledge about interested speculators from the start. Some speculators choose to wait with their ﬁrst bid until later in the auction to see the price development and patterns of other speculators. When the auction is done and the seller has accepted the last bid, the real estate agent saves all the bids for the auction and gives a complete list to the seller and the buyer.

3.2 Auction rules Auctions consist of two different models, open and closed bidding. During the open bidding process, the speculator gives the real estate agent their bid on 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 or make a new bid, which open possibilities for a bidding race between different speculators. During a closed bidding process, the speculators gets a time frame when they can give their bid on the object, making the bidding process a one-shot opportunity. When the deadline is closed, the real estate agent gives the bids to the seller who then decides which bid and speculator is more convenient to sell to. The difference here from the open bidding process is that the speculators don not know about the bids of the others. The speculator has no obligation to get any information from the real estate agent about the other speculators. When the winning speculator and the seller writes the contract, the real estate agent will give them the bidding list with information about the speculators and their bids. According to the Swedish website maklarsamfundet.se, the most usual model is the open bidding, and we will use this model in the research.

In the end of the auction process, the seller always decides who will win the action. Even though it is normally the highest bidder that wins the auction, 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 does not seem reliable and serious about the agreements or that the bidder is making the sales process take longer time than needed.

15 3.3 Bidding strategies Looking at the housing market, there are several strategies to choose from when it comes to bidding. During the auction event, it is possible to change the strategy and the strategy chosen from the start might not be the actual strategy played. Factors that inﬂuence the strategies are many. Some of the factors include how many bidders there are in the auction and how they bid, what knowledge they have or previous experience from auctions and how much money the bidder is willing to spend depending on the budget and loan constraints from the bank. Getting excited and emotionally involved with early attachment to the object could also inﬂuence the bidding and the strategy. This is a frequently occurring factor that is not rational from a game theoretical perspective.

Common strategies in the Swedish housing market Asa˚ Larsson, a real estate broker at the Swedish real estate agency Fastighets- byr˚an,Lysekil, stated in the broker blog [10] some of the more common strategies used in the Swedish housing market.

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

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

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

Waiting out the time. Before bidding, ask the broker if you can sleep on it over the weekend. With the extra time, the other opponents might ﬁnish other auctions instead, leaving fewer opponents in your auction. This could potentially lead to a reversed eﬀect, attracting new bidders to the auction, or the seller is not willing to wait.

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

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

Reacting quickly. Make a new bid as soon as there is movement in the auction. This shows all the parties involved that you’re interested and not leaving the auction.

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

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

17 4 The auction game

In this section we have constructed an auction game in MatLab to present different strategies in the Swedish housing market. The auction game makes it possible to see how the different strategies could interact and if there is a strategy that is more efficient than others. We have found connections and applied 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 strategy information from Asa˚ Larsson, and from our own researches we construct the rules 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 no knowledge about the other players in the game. The only thing they know about the other players is the number code that they have, and when they are adding their bids. Moreover, the players know their own value of the object, 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 lowest possible equilibrium. By using a certain strategy and bidding a value that is higher than the current value and lower than the budget function, the players will 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 or mixed. The game is played randomly and the bidding player will be selected randomly (no type of selected rotation between the players). The player can choose to bid below, on, or above the opening price of the object. The players in an on going game must bid above the current, existing bid, but the player can also choose to wait until other players make a bid or else the player can choose to exit the game. The auction ends when no player is bidding above the existing bid.

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

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

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

Strategy 2: Laying low The pure strategy is bidding either 5 000 or 10 000. If the player makes it to the end and there is only one competitor left, the player will bid every time a new bid is made until budget is reached.

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

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

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

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

4.2 The game The game is made from the strategies that have been found and analyzed by the real estate broker Asa˚ Larsson [10]. The game is constructed in the way that the players are rational, and stick to either a mixed or pure strategy throughout the game. Each strategy is assigned to one player. The strategies that have been used in the game are; putting everything on one card, laying low, going out strong, reacting quickly, making ’irrational’ moves, and start bidding late. The strategies are mixed or pure depending on the outline of the strategy, though in the game representation, some of the pure strategies are constructed like mixed strategies, as if the player has chosen to play so to make it more realistic.

The diﬀerent strategies have individual budgets, which is also their real value of 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

19 is 1.1 times the amount of the starting bid. The highest capital a player can get is 1.4 times the amount of the starting bid. These values come from the analysis of bidding lists that we got from diﬀerent real estate agencies in Stockholm and Gothenburg. When the budgets have been randomly distributed to the players, they will be randomly selected to start and who will play 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 and each strategy represents one player. By these conditions we wanted to see how they would behave against each others and which of them that would win the most games. First we gave them randomly distributed budgets and a 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 the strategies is better than the other. What we can see is that strategy 3 wins slightly less than the others and that strategy 1 wins slightly more. These results are pretty obvious, since for all the games, the strategy that has been distributed with the highest budget will win the game.

Figure 2: Randomly distributed budgets

20 We also wanted to see how the strategies would behave if they got the same ﬁxed budget. This time we also run the game 1 000 000 times. As we can see in ﬁgure 3, strategy 1 wins the majority of the games. But that is also not so surprising since strategy 1 is a very aggressive strategy and since they all have the same budget and strategy 1 has the strategy to place his maximum budget, it is natural that he will win and therefore a Nash equilibrium. We can also see that strategy 6 is never winning. That is because he has the strategy to start bidding late, and since strategy 1 is so aggressive, he will never 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 of them is paying less in average for an object than the others? When running the game randomly we can see (and understand from the previous results from the randomly played game) that they are paying in average about the same.

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

Strategy 4 Strategy 5 Strategy 6 3 569 900 3 540 800 3 564 700

Table 1: Average price paid from randomly played game

21 If we take the average of the results in table 1, we can see more clearly the diﬀerences 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 same amount 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 more in average than the other strategies is because his strategy is to go out strong by putting high bids and continue to do so throughout the game, which in the 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 always show the amount that we have chose the budget to be and is not providing any useful information.

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

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

Opening price: 2 405 000

Strategy(Player): Bid: 4 2 430 000 1 2 620 000 4 2 640 000 1 2 650 000 4 2 675 000 3 2 725 000 2 2 735 000 5 2 755 000 3 2 855 000 4 2 860 000 5 2 900 000 4 2 925 000 3 2 975 000 2 2 985 000 5 3 065 000 3 3 115 000 5 3 125 000 3 3 225 000

Table 2: Bidding list with random budgets

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

23 6 Conclusion

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

When we built the game we had the aim to have 10 different strategies and 15 different players that could choose from these strategies, though during the time we encountered some obstacles. Some of the strategies could not be programmed randomly, but needed to get priority before the others, and that would not make the game random as we wanted. We also wanted the the players to meet opponents with the same strategy, but there were also the case that they needed to get priority. So our final game became to include 6 strategies which all became an individual player.

From our game we got the results that there are a difference between the fixed and the random played game. Our result showed that strategy 1 was the strategy that had the most winning auctions. In the fixed budget, strategy 1 had more than 60% of the winning auctions. But these results are pretty obvious. In the game with fixed budgets it is obvious that strategy have the most wins since he will aways place his maximum budget which is the other players budgets also, and they cannot place a higher bid, and he will then win. In the game played with randomly distributed budgets it is also 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 strategy 1 has) the best strategy is to bid your true value of the object. He states that the most common winning strategy in ascending auctions is to bid the true value of the object, which here is representing the budget. Since strategy 1 is very straight forward in the game with a pure strategy set of only two strategies, bidding 30 000 less than budget, and bidding the budget value, it is clearly an aggressive strategy for winning when the budget is ﬁxed. Since the rest of the strategies are not as aggressive with their strategies and bidding values, their chance of winning is less than for strategy 1.

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

24 could not include in our strategies. One factor that we could not include was that bidders can change their strategies during the game after seeing how the other bidders behave. In our game, the information is ﬁxed from the start, which means that new information does not aﬀect the game, the players will follow the game as if it is a script. Other factors include that the bidders can make bids before the bidding game has started, bidding on more than one object, crossing borders and ask the real estate agent to sleep on it over the weekend to get more time. Another factor is that a bidder with higher 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. If we would have included these factors in our programming it would not have been a fair game, since we would be needed to give some strategies priority before others. We wanted the game to be as fair as possible and chose then to exclude these types of strategies.

Limitations Some limitations that we encountered during our research was that due to secrecy we could not interview bidders from the bidding lists that we got from the real estate agents. Since we could not interview the bidders, we did not have any information about their budgets and strategies. The limited information could not give us answers about why they dropped out from the auction. Maybe they reached their budget and if not, it would have been useful to know the reason they chose to leave the bidding anyway. From a rational perspective, they could have chosen to leave due to lack of interest or a winning object in another auction, but it could also have been from the stressful mind games that happens in auctions that aﬀect the bidders behaviours. It would had been interesting to interview the bidders what their thoughts and strategies were during their bidding. For future research in this subject, to really understand and perhaps ﬁnd out what strategy is a truly winning strategy, is to interview people. To go to open houses and ask what their budgets are and what strategy they would use to win the object and follow the auctions with this insight information to be able to fully analyze from a game theoretic perspective.

25 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: Cambridge University Press, 209-210

[3] Ge¸ckil,I K., Anderson, P. L. (2010) Applied game theory and strategic behavior. 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 of Consciousness Studies Vol. 7 (1-2) Exeter, UK: Imprint Academic

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

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

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

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

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

26 [10] Larsson, A.˚ (2013) Budgivningsstrategier. M¨aklarvardag [Blog] Available at: http://maklarvardag.blogspot.se/2013/05/budgivningsstrategi.html [Accessed 3 Apr. 2016]

[11] McKenzie, A. J. (2009) Evolutionary Game Theory. The Stanford 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 to Game 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 property development 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

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27 A Appendix: MATLAB codes

A.1 The auction

1 c l c

2 c l e a r all

3 rounds=1;

4 win=[0 0 0 0 0 0];

5 Average win=[0 0 0 0 0 0];

6 name={ ’Strategy1’,’Strategy2’,’Strategy3’,’ Strategy4’,’Strategy5’,’Strategy6’ } ;

8 while rounds ˜= 100001

9 %%%%%%%%%%%%%%%%%%

10 % starting b i d= 500;

11 %s c a p i t a l= [1000 1000 1000 1000 1000 1000];

12 %%%%%%%%%%%%%%%%%%

13 s t a r t i n g b i d= round(randi([400 5000],1,1)/5) ∗5;

14 % The lowest amount of capitala person can have:

15 m i n c a p i t a l= round(1.1 ∗ s t a r t i n g b i d);

16 % The highest amount of capitala person can have:

17 max capital= round(1.4 ∗ s t a r t i n g b i d);

18 % randomly distributes capital to each bidder

19 % between min and max of aviable capital

20 s c a p i t a l= round(randi([min c a p i t a l max capital],1,6) /5) ∗5;

22 %%%%%%%%%%%%%%%%%%

23 b i d s a r r a y= zeros(1,7);

24 competitors=6;

25 n =0;

27 o l d b i d=starting b i d;

28 auctions=1;

32 while competitors ˜= 1

33 competitors=competitors l e f t(s c a p i t a l);

34 switchn

35 case1

28 36 i f(bids 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 [ current b i d,s c a p i t a l(n),bids a r r a y]= s1(n , current bid,s c a p i t a l(n), bids array, competitors);

38 e l s e

39 end

41 case2

42 i f(bids 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 ˜= starting b i d)

43 [ current b i d,s c a p i t a l(n),bids a r r a y]= s2(n , current bid,s c a p i t a l(n), bids array, competitors);

45 e l s e

46 end

47 case3

48 i f(bids 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

49 [ current b i d,s c a p i t a l(n),bids a r r a y]= s3(n , current bid,s c a p i t a l(n), bids array, competitors);

51 e l s e

52 end

53 case4

54 i f(bids 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

55 [ current b i d,s c a p i t a l(n),bids a r r a y]= s4(n , current bid,s c a p i t a l(n), bids array, competitors);

57 e l s e

58 end

59 case5

60 i f(bids 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

61 [ current b i d,s c a p i t a l(n),bids a r r a y]= s5(n , current bid,s c a p i t a l(n), bids array, competitors);

29 62

63 e l s e

64 end

65 case6

66 i f bids a r r a y(n) ˜= 0 &&s c a p i t a l(n) ˜= 0

67 [ current b i d,s c a p i t a l(n),bids a r r a y]= s6(n , current bid,s c a p i t a l(n), bids array, competitors);

69 e l s e

70 end

71 otherwise

72 c u r r e n t b i d=starting b i d;

73 end

75 %%%%%%%%%%%%%%%%%%

76 % Auction list

77 i f current b i d˜=old b i d && rounds==100000

78 b i d l i s t(auctions ,1)=n;

79 b i d l i s t(auctions ,2)=current b i d;

80 o l d b i d=current b i d;

81 % checking the starting capital on the round

82 i f auctions==1

83 A u c t i o n startingbid=starting b i d;

84 startingcapital=repmat(s c a p i t a l ,1);

85 end

86 auctions=auctions+1;

87 end

88 %%%%%%%%%%%%%%%%%%

89 n= randi(6);

91 end

92 [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 end

96 Average win=Average win./win;

98 pie(win)

99 legend(name(win >0) )

30 A.2 Bid counter

1 f u n c t i o n[ bids a r r a y]= bid counter(n, bids a r r a y)

2 % This function isa bid counter where the first

3 % six digits of the array is counting the

4 % individual bids done by others since the last

5 % time the competitor dida bid. The seventh digit

6 % is the total sum of bids done in the game.

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

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

10 b i d s a r r a y(3)= bids a r r a y(3)+1;

11 b i d s a r r a y(4)= bids a r r a y(4)+1;

12 b i d s a r r a y(5)= bids a r r a y(5)+1;

13 b i d s a r r a y(6)= bids a r r a y(6)+1;

14 b i d s a r r a y(7)= bids a r r a y(7)+1;

16 switchn

17 case1

18 b i d s a r r a y(1)=0;

19 case2

20 b i d s a r r a y(2)=0;

21 case3

22 b i d s a r r a y(3)=0;

23 case4

24 b i d s a r r a y(4)=0;

25 case5

26 b i d s a r r a y(5)=0;

27 case6

28 b i d s a r r a y(6)=0;

29 case7

30 b i d s a r r a y(7)=0;

31 end

31 A.3 Competitors left

1 f u n c t i o n[competitors]= competitors l e f t(s c a p i t a l)

2 competitors out=0;

3 f o ri=1:length(s c a p i t a l)

4 i f 0==s c a p i t a l(i)

5 competitors out= competitors out+1;

6 end

7 end

8 competitors=6−competitors out;

9 end A.4 Winner

1 f u n c t i o n[win] = winner(s c a p i t a l,win)

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

5 end A.5 Average

1 f u n c t i o n[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 whiles c a p i t a l(i )==0

6 i=i+1;

7 end

8 s c a p i t a l(i)=current b i d;

9 Average win=Average win+s c a p i t a l;

11 end

32 A.6 Strategy 1

1 f u n c t i o n[ output bid,s c a p i t a l,bids a r r a y]= s1(n, current bid,s c a p i t a l, bids array, competitors)

2 % Strategy 1: Putting everything on one card

3 %(Pure strategy)

4 % Put max(budget) −30 000. 30 000 (30K)

5 % is based on possibility to play one more round

6 % if other players respond to the new auction price.

7 % Response: If next bid is lower than 30K from the previous,

8 % bid again up to the budget price.

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

11 output bid=s c a p i t a l;

12 b i d s a r r a y= bid counter(n,bids a r r a y);

13 e l s e i f current b i d+30 < s c a p i t a l;

14 output bid=s c a p i t a l − 30;

15 b i d s a r r a y= bid counter(n,bids a r r a y);

16 e l s e i f current b i d >=s c a p i t a l

17 s c a p i t a l=0;

18 output bid= current b i d;

19 e l s e

20 output bid=current b i d;

23 end

24 end

33 A.7 Strategy 2

1 f u n c t i o n[ output bid,s c a p i t a l,bids a r r a y]= s2(n, current bid,s c a p i t a l, bids array, competitors)

2 % Strategy 2: Laying low(Pure strategy)

3 % After the first rounds, start bidding5K −10K

4 % to make the bidding go slower and making people

5 % nervous to get to the end.

6 % Maybe other players will win another auction while

7 % still bidding on the one happening now.

8 % Response: Wait for2 players to bid before

9 % makinga new bid (10K). If there is only one player

10 % left, bid every timea new auction bid has been made . ( 5K)

13 a v a i a b l e b i d s=[5 10];

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

17 output bid= current b i d+avaiable b i d s(2);

18 b i d s a r r a y= bid counter(n,bids a r r a y);

20 e l s e i f competitors==2 &&s c a p i t a l >= current b i d+ a v a i a b l e b i d s(1);

21 output bid= current b i d+avaiable b i d s(1);

22 b i d s a r r a y= bid counter(n,bids a r r a y);

24 e l s e i f current 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 bid= current b i d;

27 e l s e

28 output bid=current b i d;

30 end

32 end

34 A.8 Strategy 3

1 f u n c t i o n[ output bid,s c a p i t a l,bids a r r a y]= s3(n, current bid,s c a p i t a l, bids array, competitors)

2 % Strategy 3: Going out strong(Mixed strategy)

3 % Making high bids though out the auction,

4 % making people scared of entering? 50K −100K.

5 % Response: Makea probability distribution 1/2

6 % for playing every timea new bid has arrived

7 % and every second new bid has arrived.

8 % If there is only one player left,

9 % bid every timea new auction bid has been made.

11 a v i a b l e b i d s=[50 100];

14 i f current b i d < s c a p i t a l &&s c a p i t a l >= current b i d +aviable b i d s(1) && competitors >=2;

15 i f bids a r r a y (3)==1

16 bid= randi(2) −1;

17 i f bid==1 &&s c a p i t a l >= current b i d+ a v i a b l e b i d s(2)

19 output bid= current b i d+aviable b i d s(randi (2) ) ;

20 b i d s a r r a y= bid counter(n,bids a r r a y);

21 e l s e i f bid==1 &&s c a p i t a l >= current b i d+ a v i a b l e b i d s(1)

22 output bid= current b i d+aviable b i d s(1);

23 b i d s a r r a y= bid counter(n,bids a r r a y);

25 e l s e

27 output bid=current b i d;

29 end

30 e l s e

31 output bid= current b i d+aviable b i d s(randi(2));

32 b i d s a r r a y= bid counter(n,bids a r r a y);

33 end

35 35

36 e l s e i fs c a p i t a l >= current b i d+aviable b i d s(2) && competitors==2;

37 output bid= current b i d+aviable b i d s(randi(2));

38 b i d s a r r a y= bid counter(n,bids a r r a y);

39 e l s e i fs c a p i t a l >= current b i d+aviable b i d s(1) && competitors==2;

40 output bid= current b i d+aviable b i d s(1);

41 b i d s a r r a y= bid counter(n,bids a r r a y);

42 e l s e i f current 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 bid= current b i d;

45 e l s e

46 output bid=current b i d;

49 end

52 end

36 A.9 Strategy 4

1 f u n c t i o n[ output bid,s c a p i t a l,bids a r r a y]= s4(n, current bid,s c a p i t a l, bids array, competitors)

2 % Strategy 4: Reacting quickly(Mixed strategy)

3 % Makinga new bid as soon as another bid enters

4 % witha probability distribution over5K − 30K.

5 i=6;

6 whiles c a p i t a l <= current b i d+i ∗5

7 i fs c a p i t a l >= current b i d+i ∗5

8 break

9 e l s e i fi==1

10 break

11 e l s e

12 i=i −1;

13 end

14 end

15 p l a c i n g b i d=current b i d+randi(i) ∗5;

17 i f current b i d >=s c a p i t a l −5

18 s c a p i t a l=0;

19 output bid= current b i d;

20 e l s e i f current 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 ∗5

21 output bid= current b i d+randi(i) ∗5;

22 b i d s a r r a y= bid counter(n,bids a r r a y);

24 e l s e

25 output bid=current b i d;

27 end

37 A.10 Strategy 5

1 f u n c t i o n[ output bid,s c a p i t a l,bids a r r a y]= s5(n, current bid,s c a p i t a l, bids array, competitors)

2 % Stratey 6: Making irrational moves

3 %(Mixed strategy)

4 % Making uneven bids witha probability

5 % distribution overa set of numerical

6 % values to make changes in the auction

7 % and make other players confused

8 % with their own bidding strategies.

9 %5K −100K depending on budget and time of game.

11 i=20;

12 whiles c a p i t a l <= current b i d+i ∗5

13 i fs c a p i t a l >= current b i d+i ∗5

14 break

15 e l s e i fi==1

16 break

17 e l s e

18 i=i −1;

19 end

20 end

22 p l a c i n g b i d=current b i d+randi(i) ∗5;

24 i f current b i d >=s c a p i t a l −5

25 s c a p i t a l=0;

26 output bid= current b i d;

27 e l s e i f current 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 ∗5

28 output bid= current b i d+randi(i) ∗5;

29 b i d s a r r a y= bid counter(n,bids a r r a y);

31 e l s e

32 output bid=current b i d;

33 end

38 A.11 Strategy 6

1 f u n c t i o n[output bid,s c a p i t a l,bids a r r a y]= s6(n, current bid,s c a p i t a l, bids array, competitors)

2 % Strategy 7: Start bidding late(Mixed strategy)

3 % Waiting until there is only two player left in

4 %the auction if the price is still below the

5 % budget. Probability distribution over5K −100K.

6 i=20;

7 whiles c a p i t a l <= current b i d+i ∗5

8 i fs c a p i t a l >= current b i d+i ∗5

9 break

10 e l s e i fi==1

11 break

12 e l s e

13 i=i −1;

14 end

15 end

16 p l a c i n g b i d=current b i d+randi(i) ∗5;

18 i f current b i d >=s c a p i t a l −5

19 s c a p i t a l=0;

20 output bid= current b i d;

21 e l s e i f placing b i d <=s c a p i t a l && competitors <= 3 ;

22 output bid= placing b i d;

23 b i d s a r r a y= bid counter(n,bids a r r a y);

25 e l s e

26 output bid=current b i d;

28 end