Game Theory by Information Resources Management Association

Author:Information Resources Management Association.
Language: eng
Format: epub
Publisher: IGI Global

A simple way of modeling the above situation would be to let the players make observations directly in the G situation. Each player would then observe the payoffs of the actual game plus some error terms. However, an alternative formulation is used where the selected game (g) is observed indirectly through some parameter space which is mapped on G (Carlsson, 1993).

The typical bar problem is a well-known example of global games (Krishnamurthy, 2009), (Krishnamurthy, 2008). To model the bar problem as a global game, consider the following analogy comprising of a large number n of guests and k bars. Each guest receives noisy information about the quality of music playing at each bar. Each guest i∈{1,…,n} is in one of the bars l∈{1,…,k}. Let γl denote the fraction of guests in the bar l. Let ψl denote the quality of music playing at the bar l. Each guest i obtains a noisy k-dimensional measurement vector Y(i) about the quality of music =(1,…,k) playing at bars. Based on this noisy information Y(i), each guest needs to decide whether to stay in the current bar, or to leave the bar. When a guest selects the bar, he receives a payoff based on the music quality (). If he goes from the bar l to the bar m, his payoff (Ulm) is defined as follows.

, s.t., l,m∈{1,…,k} (29)

where plm denotes the probability that if a guest leaves the bar l, he will move to the other m bar (m≠l). The payoff means that the expected quality of music he would receive from the other k – 1 bars. If a guest chooses to stay in the bar l, the payoff (Ull) for him is given by.

Ull = , s.t., l,m∈{1,…,k} (30)

where αl denotes the fraction of guests that decided to stay in the bar l, so that (αl×γl) is the fraction of all guests that decided to stay in the bar l. ξm is the probability that a guest leaves the bar m (m≠l). As far as guests in the bar l are concerned, they have limited information about how guests in the other bars decide whether to stay or leave. This limited information is represented as the probability ξm and (pml×ξm) denotes the probability that a guest leaves the bar m and goes to the bar l. Finally, denotes the fraction of all guests that move to the bar l from the other k – 1 bars. In the payoff in (30), the first function is an increasing function of , which is obtained according to the quality of music. gl(∙) function implies that the better the music quality, the higher the payoff to the guest in staying at the bar l. The second function fl(∙) is defined by using the noisy information vector Y. Typically, the function fl(∙) is quasi-concave. For example, if too few guests are in the bar l, then fl(∙) is small due to lack of social interaction. Also if too many guests are in the bar, then fl(∙) is also small due to the crowded nature of the bar.

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