"Game for each and each for game"justify the statement with reference to its objects
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Answer:
Specific games stand for significant recurrent patterns of social interaction. In a perspective called ‘logic of games’, notions and results from logic are used to analyze the structure of various games. In fact, much classic reasoning about games involves notions that are familiar from logic.
Example Game solution reasoning.
Consider the following game tree for two players A, E, with turns marked, and with pay-offs written with the value for A first. Alternatively, these values can be interpreted as encoding players’ qualitative (or ordinal) preferences between outcomes.
This is a game tree diagram illustrating what is described in the paragraph following the figure. The extended description (link in figure caption) will describe the tree.
Figure 1. ⓘ
Here is how players might reason. At her turn, E faces a standard decision problem, with two available actions and the outcome of action left better for her than that of right. So she will choose left. Knowing this, A expects that his choosing right will give him outcome 0, while going left gives him outcome 1, so he chooses left. As a result, both players are worse off than they would have been, had they played right/right. The reasoning in this scenario, in short, leads to an outcome that is not Pareto-optimal.
The example raises the question just why players should act this way, and whether, say, a more cooperative behavior could also be justified. An answer obviously depends on the players’ information and style of reasoning. Here it becomes of interest to probe the structure of the example. Looking more closely, many notions are involved in the above scenario: actions and their results, players knowledge about the structure of the game, their preferences about its results, but also how they believe the game will proceed. There are even counterfactual conditionals in the background, such as A’s explaining his choice afterwards by saying that “if I had played right, E would have played left”. These notions, moreover, are entangled in subtle ways. For instance, A does not choose left because it dominates right in the standard sense of always being better for him, but rather because left dominates right according to his beliefs. How these beliefs are formed, in turn, depends on many other features of the game, including the nature of the players.
In short, even a very simple game like the one discussed brings together large parts of the agenda of philosophical logic in one very concrete setting. This entry will zoom in on the aspects mentioned here, with Section 3 dedicated to players’ preferences and beliefs, while Section 4 addresses reasoning styles and the dynamics of attitudes as the game proceeds.
The analysis is structured by a few broad distinctions. Intuitively, games involve several phases that involve logic in different ways: deliberation prior to the game, as many game-theoretic solution concepts are in fact deliberation procedures that create initial expectations about how a game will go on. Observation and belief revision during game play, including reactions to deviations from prior expectations. And finally, post-game analysis, say, to settle what can be learnt from a defeat, or to engage in spin about one’s performance. Moreover all this can be considered in two modes, assuming either a first-person participant or a third-person observer view of games and play.