WEEK 2
1. Introduction
This chapter introduces the concept of a game and encourages the reader to begin thinking about the formal analysis of strategic situations. The chapter contains a short history of game theory, followed by a description of “non-cooperative theory” (which the book emphasizes), a discussion of the notion of contract and the related use of “cooperative theory,” and comments on the science and art of applied theoretical work. The chapter explains that the word “game” should be associated with any well-defined strategic situation, not just adversarial contests. Finally, the format and style of the book are described.

Lecture Notes
The non-administrative segment of a first lecture in game theory may run as follows.
• Definition of strategic situation.
Interdependence—Ndervartesi: sjellja e nje personi ndikon ose pozitivisht ose negativisht ne mbarevajtjen e puneve te nje personi tjeter;
Strategic setting—Situate strategjike: Situatat a ndervartesise quhen situate strategjike sepse qe nje person te vendose se cila eshte levizja e tij me e mire, ai duhet te marre parasysh se si te tjeret perreth tij do te zgjedhin levizjet e tyre;
Theory of strategic interaction—Teori e nderveprimit strategjik: duhet per shkak se, se pari, identifikon gjuhen ne mund te bisedojme dhe shkembejme ide rreth sjelljes njerezore; se dyti, na jep nje kornize qe na udhezon te ndertojme modele te situatave strategjike; se treti, na ndihmon te gjurmojme implikimet logjike te pandehmave rreth sjelljes;
Individual actions—Veprime individuale: jane ato veprime qe individed I vendosin vete, pavaresisht nga personat e tjere qe ndodhen ne ate situate strategjike;
Noncooperative game—Loje jobashkepunuese: eshte loje e luajtur nga aktore qe ndermarrin veprime individuale ne nje situate strategjike;
Join action—veprim i perbashket: keshtu quhen perfundimet e tratativave;
Cooperative game—Loje bashkepunuese: keshtu quhen lojrat ne te cilat individed qe ndodhen ne nje situate strategjike ndermarrin veprime te perbashketa; lojrat bashkepunuese perdoren per te studjuar marrdheniet kontraktuale;

• Examples (have students suggest some): chess, poker, and other parlor games; tennis, football, and other sports; firm competition, international trade, firm/employee relations, and other economic examples; biological competition; elections; and so on.
• Competition and cooperation are both strategic topics. Game theory is a general methodology for studying strategic settings (which may have elements of both competition and cooperation).
• The elements of a formal game representation.
• A few simple examples of the extensive form representation (point out the basic components).

2. The Extensive Form
This chapter introduces the basic components of the extensive form in a non-technical way. Students who learn about the extensive form at the beginning of a course are much better able to grasp the concept of a strategy than are students who are taught the normal form first. Since strategy is perhaps the most important concept in game theory, a good understanding of this concept makes a dramatic difference in each student’s ability to progress. The chapter avoids the technical details of the extensive form representation in favor of emphasizing the basic components of games. The technical details are covered in Chapter 14.

Lecture Notes
The following may serve as an outline for a lecture.
• Basic components of the extensive form: nodes, branches.
Nodes—Nyjet: nyjet jane vendet ku veprohet;
Branches—Deget: jane veprimet individuale te ndermarra nga lojtaret;
• Example of a game tree.
• Types of nodes:
Initial node—Nyje fillestare: jane nyjet ku fillon loja;
Terminal nodes—Nyje perfundimtare: jane nyjet qe perfaqesojne fundin e lojes;
Decision nodes—Nyje te vendimeve: jane nyjet ku merren vendime;
• Build trees by expanding, never converging back on themselves. At any place in a tree, you should always know exactly how you got there. Thus, the tree summarizes the strategic possibilities.
• Player and action labels. Try not to use the same label for different places where decisions are made.
• Information sets—Sete informacioni: jane vendet ku merren vendimet; nje set informacioni perbehet nga nje ose disa nyje vendimesh; ajo pershkruan se cilat nyje vendimesh lidhen me njera tjetren me nje vije te nderprere (cka do te thote qe nje lojtar nuk dallon dot mes tyre);
Start by describing the tree as a diagram that an external observer creates to map out the possible sequences of decisions. Assume the external observer sees all of the players’ actions. Then describe what it means for a player to not know what another player did. This is captured by dashed lines indicating that a player cannot distinguish between two or more nodes.
• We assume that the players know the game tree, but that a given player may not know where he is in the game when he must make any particular decision.
Payoff/utilities—utilitetet: preferencat e lojtareve per perfundimet e lojes;
Simultaneous moves—levizjet e njekoheshme: jane levizjet qe lojtaret ndermarrin njekohesisht dhe ne te cilat ata nuk e dijne ne cilin pozicion ndodhen ndaj levizjes se kundershatrit;

3. Strategies
As noted already, introducing the extensive form representation at the beginning of a course helps the students appreciate the notion of a strategy. A student that does not understand the concept of a “complete contingent plan” will fail to grasp the sophisticated logic of dynamic rationality that is so critical to much of game theory. Chapter 3 starts with the formal definition of strategy, which is then illustrated with some examples. The critical point is that strategies are more than just “plans.” A strategy prescribes an action at every information set, even those that would not be reached because of actions taken at other information sets.

Lecture Notes
The following may serve as an outline for a lecture.
• Formal definition of strategy.
Strategy—strategjia: strategjia eshte nje plan i plote i kushtezuar per nje lojtar ne loje; strategjia e nje lojtari pershkruan ate se cka ai do te beje ne seicilin nga setet e tij te informacionit;
“Nje plan i plote i kushtezuar” do te thote nje specifikim i plote i sjelljes se nje lojtari, pra pershkrimi i veprimeve qe lojtari do te ndermarre ne seicilen nga pikat e tij te vendimit.
• Examples of strategies.
• Notation: strategy space Si, individual strategy si ∈ Si. Example: Si = {H, L} and si = H. Strategy space/strategy set—Hapesira e strategjise/Seti i strategjise: pemban cdo strategji te mundeshme te nje lojtari ne nje loje;
• Refer to Appendix A for more on sets.
• Strategy profile: s ∈ S, where S = S1 × S2 × ·· ·×Sn (product set).
Strategy profile—Profili i strategjise: eshte nje vektor i strategjive, nje per cdo lojtar; profili i strategjise pershkruan strategjite e te gjithe lojtareve ne loje;
• Notation: i and −i, s = (si, s−i).
• Discuss how finite and infinite strategy spaces can be described.
• Why we need to keep track of a complete contingent plan: (1) It allows the analysis of games from any information set, (2) it facilitates exploring how a player responds to his belief about what the other players will do, and (3) it prescribes a contingency plan if a player makes a mistake.

4. The Normal Form
Building on the definition of strategy in Chapter 3, Chapter 4 describes how each strategy profile leads to a single terminal node (an outcome), via a path through the tree. This leads to the definition of a payoff function. The chapter then defines the normal form representation as comprising a set of players, strategy spaces for the players, and payoff functions. The matrix form, for two-player, finite games, is illustrated. The chapter then briefly describes seven classic normal form games. The chapter concludes with a few comments on the comparison between the normal and extensive forms.

Lecture Notes
The following may serve as an outline for a lecture.
• Describe how a strategy implies a path through the tree, leading to a terminal node and payoff vector.
• Examples of strategies and implied payoffs.
• Definition of payoff function, ui : S → R, ui(s). Refer to Appendix A for more
on functions: Per cdo lojtar i, ne mund te percaktojme nje funksion ui : S → R (nje funksion domeini i te cilit eshte seti i profileve te strategjive dhe rrezja e te cilit jane numrat reale) ne menyre te tille qe, per cdo profil strategjik s ∈ S qe lojtari zgjedh, ui(s) eshte perfitimi i lojtarit i nga loja. Funksioni ui quhet funksioni perfitues i lojtarit i.
• Example: a matrix representation of players, strategies, and payoffs. (Use any abstract game, such as the centipede game.)
• Formal definition of the normal form: Nje loje ne formen normale (quhet gjithashtu edhe forma strategjike) konsiston ne nje set lojtaresh, {1, 2, …, n}, haperiren strategjike te lojtareve, S1, S2,….Sn, dhe funksionet perfituese per lojtaret u1, u2, … un. Keto lojra mund te shprehen me matrica dhe quhen edhe lojra matricore.
• Note: The matrix representation is possible only for two-player, finite games.
Otherwise, the game must be described by sets and equations.
• The classic normal form games and some stories. Note the different strategic
issues represented: conflict, competition, coordination, cooperation.
• Comparing the normal and extensive forms (translating one to the other).
• Discussion of the art and science of game theoretic work. A game theory model helps us organize our thoughts and isolate components of a strategic situation. The key to application: find the simplest model that can yield insight.

5. Beliefs, Mixed Strategies, and Expected Utility
This chapter describes how a belief that a player has about another player’s behavior is represented as a probability distribution. It then covers the idea of a mixed strategy, which is a similar probability distribution. The appropriate notation is defined. The chapter defines expected payoff and gives some examples of how to compute it. At the end of the chapter, there are a few comments about cardinal versus ordinal utility (although it is not put in this language) and about how payoff numbers reflect preferences over uncertain outcomes. Risk preferences are discussed in Chapter 25.

Lecture Notes
The following may serve as an outline for a lecture.
• Shembull: Ne Dilemen e te Burgosurit, lojtari 1 mund te mendoje qe lojtari 2 eshte shume e mundur te luaje strategjine C. Besimi qe lojtari 1 ka qe lojtari dy do te luaje strategjine C shprehet me shifren probabilistike p. Probabiliteti qe lojtari 1 beson se lojtari 2 nuk do do te luaje strategjine C eshte p = 0. Athere, lojtari 1 beson se lojtari 2 do te luaje strategjine D eshte
1 - p. Numrat p dhe 1 - p perbejne nje shperndarje probabilistike mbi setin {C, D}.
Perkufizim: Matematikisht, besimi i lojtarit i eshte nje shperndarje probabilistike mbi strategjite e lojtareve te tjere. Ne e shenojme ate me θ-i ∈ ΔS-i ku ΔS-i eshte seti i shperndarjes probabilistike mbi strategjite e te gjithe lojtareve me perjashtim te lojtarit i.
• Translate into probability numbers.
• Other examples of probabilities.
• Verejtje: Besimi i lojtarit i rreth sjelljes se lojtarit j eshte nje funksion θ j ∈ ΔSj i tille qe, per cdo strategji sj ∈ Sj te lojtarit j, θ j(sj) interpretohet si probabiliteti qe lojtari i mendon qe lojtari j do te luaje sj. Si shperndarje probabilistike, θ j ∈ [0, 1], per cdo sj ∈ Sj , dhe Σsj∈Sj θ j(sj) = 1.
• Examples and alternative ways of denoting a probability distribution: for Sj = {L,R} and θ j ∈ Δ{L,R} defined by θ j(L) = 1/3 and θ j(R) = 2/3, we can write θ j = (1/3, 2/3).
• Mixed strategy. Notation: σi ∈ ΔSi.
Perkufizim: Nje strategji e perzier per nje lojtar eshte akti i perzgjedhjes se nje strategjie sipas nje shperndarje probabilistike. Per te shmangur konfuzionin, ndonjehere ne i quajme strategjite e rregullta strategji te pastra [pure strategy] per ti dalluar ato nga strategjite e perziera. Nje set strategjish te perziera perfshin setin e strategjive te pastra, seicila prej te cilave eshte nje strategji e perzier qe mbart ne vetvete te gjithe probabilitetin.
• Refer to Appendix A for more on probability distributions.
• Definition of expected value: Kur lojtari i ka besimin θ-i rreth strategjive te te tjereve dhe planifikon te zgjedhe strategjine si, atehere perfitimi i priteshem eshte perfitimi mesatar qe do te merret nese ai do te luante strategjine si dhe te tjeret do te luanin sipas θ-i.

ui(si, θ-i) = Σθ-i(s-i)ui(si, s-i)

• Examples: computing expected payoffs.• Briefly discuss how payoff numbers represent preferences over random outcomes, risk. Defer elaboration until later.