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Oyun Teorisi
Kaynakların kıt olduğu bir ortamda amaçlarını gerçeklemeye çalışan iki ya da daha fazla sayıda karar verici rekabet halindedirler. Diğer bir deyişle kaynakları paylaşım çabası içindedirler. Karar vericilerin bu paylaşımda kendilerine en yüksek getiriyi sağlamak için birbirlerine karşı kullandıkları stratejileri vardır ve bu stratejileri mümkün olan en akılcı şekilde kullanırlar.

Karar vericiler varsa, karar vericiler stratejilere sahiplerse, karar vericilerin stratejilerinin sayısal değerleri ölçülebiliyorsa ve karar vericiler her şartta akılcı hareket ediyorlarsa o halde karar vericiler arasındaki rekabet problemi matematiksel olarak modellenebilir ve çözülebilir. 1944 yılında Neumann ve Morgenstern bu rekabet problemini rekabetçi (0 toplamlı) ve işbirlikçi durumlara göre formüle etmişler ve geliştirdikleri yönteme de Oyun Teorisi adını vermişlerdir. Daha sonra 1954 yılında Nash, hem rekabetçi hem de işbirlikçi oyunlarda kullanılabilecek bir denge kavramını ortaya çıkarmıştır.
Oyun Teorisi, belirli bir hedefe yönelik karar verme gücüne sahip birimlerden oluşan sistemleri incelemekte kullanılan matematiksel bir yöntemdir (Sanver, 2002).

Oyun Teorisi Terminolojisi ve Varsayımları
Aşağıda Oyun Teorisinde kullanılan bazı temel kavramlar ve varsayımlar açıklanmıştır.
Oyuncular: Bir oyunda amaçlarını optimize etmeye çalışan kişi ya da kurumlar. Oyunda en az iki oyuncu bulunur ve akılcı hareket ettikleri gibi, kazanmak için en iyisini yaptıkları varsayılır.
Stratejiler: Her oyuncunun sahip oldukları eylem seçenekleri. Bir oyuncu için herhangi bir strateji kural olup, seçenekler oyunun seçimini belirler. Herhangi bir oyuncunun seçenekleri belirsiz sayıdaysa oyun sonlu değil süreklidir. Seçenek sayısı belirli ise oyun sonludur
Kazanç veya Ödemeler: Oyunun sonucu kazanma, yitirme veya oyundan çekilme olabilir. Her sonuç veya ödeme, negatif, pozitif veya sıfır olmak üzere her oyuncunun rakibine karşı kazancını veya kaybını belirler.
Ödemeler Matrisi: Bu matris, oyuncuların strateji seçimlerinin türlü bileşiminden sonuçlanan kazanç veya kayıpları gösterir. Ödeme matrisinin elemanları pozitif, negatif veya sıfıra eşit olabilir. Matrisin herhangi bir elemanı pozitif ise sütunda yer alan oyuncu, satırda yer alan oyuncuya bu miktarda ödeme yapar. Matrisin herhangi bir elemanı negatif ise satırdaki oyuncu, sütundaki oyuncuya bu negatif elemanın mutlak değerine eşit ödemede bulunur. Matrisin elemanı sıfır ise oyunculardan hiçbiri birbirine ödemede bulunmaz. Ödemeler matrisi sadece bir oyuncunun değerlerini temsil eder.
Oyunlar: Oyunların sınıflandırılması genellikle oyuncuların sayılarına göre yapılır. İki kişilik, üç kişilik veya (n) kişilik oyunlar kurulabilir. n=2 ise oyun 2 kişilik, n≥2 ise oyun n kişili oyundur. Ayrıca sıfır toplamlı, sabit toplamlı olmayan ve sıfır toplamlı olmayan oyunlar olarak da oyunlar sınıflandırılır.
Tam (arı) Stratejiler: Oyunun sonucunu tek bir strateji çiftinin oluşturması durumu. Söz konusu sonuç her oyuncu için olabilecek en iyi sonuçtur. Tam stratejiler, oyunun tepe (eyer) noktasını belirler.
Karma Stratejiler: Oyunun sonucunu birden fazla strateji çiftinin belirlemesi durumu. Strateji çiftleri olasılık değerleri ile ifade edilir ve oyunun sonucunu oluşturan strateji çiftleri olasılık değerleri toplamı 1 dir.
Beklenen Değer: Oyunun sonucunda herhangi bir oyuncunun elde edeceği değer. Beklenen değer strateji çiftlerinin gerçekleşme olasılıkları ile değerlerinin çarpımlarının toplamıdır.
Oyun Teorisinin Temel Mantığı
Oyunun sonucu ister arı strateji ister karma strateji olsun çözüm süreci ödemeler matrisi üzerinde gerçekleştirilir. Çözüm süreci oyunun hangi oyuncu açısından değerlendirileceğinin seçimi ile başlar. Eğer ödemeler matrisinin satırlarını temsil eden oyuncu için çözüm gerçekleştirilecekse maximin (minimumların maksimumu) yöntemi, sütunlarını temsil eden oyuncu için çözüm gerçekleştirilecekse minimax (maksimumların minimumu) yöntemi uygulanır. Oyunun sonucunda maximin ve minimax değerleri birbirine eşitse, oyun arı stratejili bir oyundur.

Maximin yönteminde öncelikle ödemeler matrisinin her bir satırının en küçük elemanı seçilir. Daha sonra bu değerler arasından en büyüğü belirlenir. Bulunan değer ödemeler matrisinde satırları temsil eden oyuncunun beklenen değeridir. Çünkü oyuncu satırlardaki büyük değerin seçilmesi durumunun diğer oyuncu tarafından tercih edilmeyeceğini ve diğer oyuncunun oyunu terk edeceğini bilir. Bu oyuncu açısından en küçük değerlerin en büyüğü ise mantıklı bir sonuç olacaktır. Diğer bir deyişle bu oyuncu açısından geçerli strateji kötülerin iyisi olarak özetlenebilir.

Sütunları temsil eden oyuncu açısından bakıldığında ise bu kez doğru mantık iyilerin kötüsü olacaktır. Çünkü sütunları temsil eden oyuncu diğer oyuncunun maximin stratejisini bilir ve oyunu minmax stratejisi ile oynar. Sütunları temsil eden oyuncu elemanlarını gözden geçirir ve her bir sütunun en büyük değerini seçer. Bu oyuncu açısından oyunun sonucu bu değerlerin en küçüğüdür (Taha, 1997).

Örnek
Rekabet halindeki A ve B firmalarından yıllık kar (milyon $) açısından A’ nın 3, B’ nin ise 4 stratejisi bulunmaktadır. A firmasına göre düzenlenen ödemeler matrisi aşağıda gösterilmiştir. Buna göre A ve B firmaları arasındaki rekabet oyununu değerlendiriniz.

| | |B |
| | |B1 |B2 |B3 |B4 |
| |A1 |6 |- 1 |5 |3 |
|A | | | | | |
| |A2 |10 |4 |5 |- 3 |
| |A3 |6 |5 |7 |4 |

A oyuncusu maximin mantığı ile hareket edecek ve B oyuncusunu oyunda tutmak için kendi stratejilerini temsil eden satır değerlerinin en küçüklerini seçecektir. Bu değerler arasından en büyüğü olan 4 değeri ise A oyuncusu için en iyi değerdir. Diğer bir deyişle A’ nın en iyi stratejisi A3 stratejisidir.

| | |B |
| | |B1 |B2 |B3 |B4 |
| |A1 |6 |- 1 |5 |3 |- 1 | |
|A | | | | | | | |
| |A2 |10 |4 |5 |- 3 |- 3 | |
| |A3 |6 |5 |7 |4 |4 |maximin |
| | |10 |5 |7 |4 | | |

minimax

A’ nın strateji mantığını bilen B oyuncusu ise minimax mantığı ile hareket edecek ve öncelikle kendi stratejilerini temsil eden sütun değerlerinin en büyüklerini seçecektir. Minimax mantığına göre B oyuncusunun geçerli stratejisi, bu değerler arasından en küçüğünü seçmek olacaktır. Yukarıdaki örneğe göre bu değer 4 yani B4 stratejisidir.

Sonuçta maximin ve minmax değerleri birbirine eşit olduğundan bu oyun arı stratejili yani tepe noktalı bir oyundur ve oyunun sonucunda A oyuncusunun beklenen değeri 4 (milyon $) olarak gerçekleşecektir. Bu değer ise B oyuncusu açısından bir kayıp olacaktır.

Karma Stratejili Oyunlar ve Çözüm Yöntemleri
Karma stratejili oyunların belirgin özelliği, ödemeler matrisindeki maximin ve minimax değerlerinin birbirine eşit olmamasıdır. Bu durum ise oyunun sonucunun tek bir strateji çifti olmaması anlamına gelir. Aşağıda karma stratejili oyunlar için kullanılabilecek Grafik Yöntem ve Doğrusal Programlama Yaklaşımı açıklanmıştır.

a) Grafik Yöntem
Eğer ödemeler matrisi B oyuncusu açısından [pic]) ya da A oyuncusu açısından [pic] boyut şartlarından birini taşıyorsa ya da ödemeler matrisi matris işlemleriyle bu boyutlara indirgenebiliyorsa, oyun Grafik Yöntemle çözülebilir. Diğer deyişle satır ya da sütunları temsil eden oyunculardan biri 2’ den fazla stratejiye sahip olmamalıdır. Burada Grafik Yöntem, satırları temsil eden oyuncunun (A oyuncusu) iki stratejiye sahip olması durumuna göre anlatılmıştır.

Koordinat sisteminin yatay ekseni 2 stratejiye sahip oyuncunun 1. stratejisinin gerçekleşme olasılığını [pic] gösterir. Söz konusu olasılık değeri doğal olarak [pic] aralığında olacaktır. Bu durumda oyuncunun 2. stratejisinin olasılık değeri [pic] olacaktır.

Daha sonra A oyuncusunun, B oyuncusunun stratejileri [pic] karşısındaki beklenen değerleri [pic] hesaplanır. Beklenen değer,[pic] satır vektörü ile ödemeler matrisindeki B oyuncusunun ilgili stratejilerine karşılık gelen sütun vektörlerinin çarpımına eşittir. Diğer bir deyişle A oyuncusuna ilişkin ödemeler matrisi,

[pic]

şeklinde ise beklenen değer (3.1) formülü ile hesaplanabilir.

[pic] (3.1)

Görüldüğü gibi beklenen değerler doğru denklemi formatındadır. Daha sonra elde edilen doğru denklemleri grafik eksene işlenir. Koordinat sisteminin düşey ekseni beklenen değerleri gösterir. Koordinat sisteminin [pic] ve [pic] için iki düşey ekseni vardır.

Koordinat sistemindeki mümkün çözüm noktaları doğruların kesiştiği noktalarda gerçekleşir. A oyuncusunun maximin yöntemine göre hareket ettiği göz önüne alındığında mümkün noktalardan optimal olanı, minimumların maksimumunda gerçekleşenidir.

Örnek
A oyuncusunun ödemeler matrisi,
[pic]
ise, bu oyunun sonucunu bulunuz.

Bu oyunda A oyuncusunun 2 stratejisine karşılık B oyuncusunun 4 stratejisi bulunmaktadır. B oyuncusunun stratejilerine karşılık A oyuncusunun beklenen değerleri (3.1) formülünden aşağıdaki gibi hesaplanmıştır.

[pic]
[pic]
[pic]
[pic]

Bu doğruların koordinat sisteminin düşey eksenlerini kestiği noktalar ise,
[pic] için [pic]

[pic] için [pic]

[pic] için [pic]

[pic] için [pic]

olarak bulunur. Söz konusu doğrular aşağıdaki şekilde gösterilmiştir.

Şekil 3.1: Grafik Çözüm
| |E(A) | | | | | | |
|AK |TD |[pic] |[pic] |[pic] |[pic] |[pic] |Çözüm |
|0 |[pic] |1 |3 |2 |1 |0 |1 |
|0 |[pic] |2 |1 |3 |0 |1 |1 |
| |[pic] |0 |0 |0 |0 |0 |0 |
| |[pic] |1 |1 |1 |0 |0 | |

Simplex yöntem 2 iterasyon sürdürüldüğünde ise optimal tabloya ulaşılır. Optimal tablo aşağıda gösterilmiştir.

| |[pic] |1 |1 |1 |0 |0 | |
|AK |TD |[pic] |[pic] |[pic] |[pic] |[pic] |Çözüm |
|1 |[pic] |0 |1 |1/5 |2/5 |-1/5 |1/5 |
|1 |[pic] |1 |0 |7/5 |-1/5 |3/5 |2/5 |
| |[pic] |1 |1 |8/5 |1/5 |2/5 |3/5 |
| |[pic] |0 |0 |-3/5 |-1/5 |-2/5 | |

Optimal tablodan görüleceği gibi B oyuncusu [pic] ve [pic] stratejilerini kullanacak, [pic] stratejisini ise hiç kullanmayacaktır. Bu stratejilerin gerçek değerleri ise, [pic] olduğu dikkate alınarak [pic] formülü yardımıyla bulunabilir. Bu durumda,

[pic]ve [pic] olarak bulunur. Diğer bir deyişle B oyuncusunun [pic] stratejilerinin olasılık değerleri [pic] dır. Yukarıda anlatıldığı gibi [pic]sırasından ise A oyuncusunun [pic] stratejilerinin olasılık değerleri [pic] olarak elde edilir.

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...Module-16 The role of business in The economy: M arkets and coMMerce TEACHER’S GUIDE P. 487 P. 491 P. 492 P. 492 P. 493 P. 496 P. 497 P. 501 Defined Content standards Materials Procedure Lesson outline Closure Assessment Overheads VISUAlS N Visuals for overhead projector. Copy to transparent paper for overhead. P. 502 NVisual-1: Market defined P. 503 NVisual-2: Business defined lESSonS 2 Copy and handout to students. P. 506 P. 513 P. 518 P. 522 2Lesson-I: I, Pencil 2Lesson-II: Specializing in production 2Lesson-III: Wal-Mart 2Lesson assessment the role of business in the econoMy Markets and coMMerce Module-16 Teacher DEFInED A market is a social arrangement that allows buyers and sellers to discover information and complete voluntary exchanges of goods and services. commerce consists of trading something of economic value such as a good, service, information, or money between two entities. business is the management and coordination of people and resources to accomplish particular production goals, usually for the purpose of making profit. business, commerce, and markets create many benefits for consumers, and as a result of business, commerce, and markets in the economy, consumers experience an increased standard of living. businesses obtain resources such as labor and equipment, and businesses specialize in the production of particular goods and services. As a result of business in the economy, the variety...

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...offer the bases in preparing, presenting and displaying even in interpreting general-purpose financial statements. There are some important accounting concepts that support the readiness and preparation of any accounting arrangement or financial statements such as Going Concern Concept, Consistency Concept, Prudence Concept, and Accruals Concepts. For example, Going Concern Concept is a company or organizations will not be going to bankrupt unless there have a confirmation and evidence to the contrary and this is supported by accountants. Purpose of Accounting Conventions An accounting convention refer to regular and common practices which are all around followed in recording and exhibiting accounting data and information of the business entity. They are taken after like traditions, convention, and so forth in a general public. Accounting conventions are evolved through the normal and consistent practice throughout the years to encourage uniform recording in the books of accounts. Accounting conventions help in contrasting accounting information of various specialty units or of the same unit for various...

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