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Say if there is a team of gamblers betting on stock exchange. We can model the outcome of stock exchange as state of nature, because it is not deterministic. So the objective of team of gamblers is to maximize their collective profit so would they be playing Bayesian game or Stochastic game?

Since we cant predict (the information of state of nature is hidden) the gamblers may also not know what strategy their peers are playing if their peers are busy monitoring live feed and betting. So what is such game called?

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To me, Bayesian games are usually "static" in the sense that no "time" is involved. The state is drawn once, and players make decisions, possibly sequentially, and the outcome is determined --- that's the end of the game.

On the other hand, stochastic games are "dynamic" in the sense that states are drawn repeatedly, and after each draw, players make decisions (again, possibly sequentially), which form a part of the history of the game.

In your case, if each gambler makes only one trade, say on day $1$, and does not touch their portfolio until day $30$ when they cash out completely, then I'd call the game Bayesian. However, since it's more likely that the gamblers would each make multiple trades on different days while they remain in the market, as anyone actively "betting" on stock markets would, the game they play would be dynamic and thus stochastic game would be a better label.

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  • $\begingroup$ Ok it makes sense but remember the gamblers also analysing each others strategy, i.e. eventhought they are working together, due to too much work load they are not talking to each other, so in a way their strategy is hidden, hence, if each of them also have to consider what others "might" be betteing on. Does this imply its Bayesian game regardless, perhaps dynamics could be somehow accomodated? $\endgroup$
    – user0193
    Commented Jan 2, 2022 at 8:29
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    $\begingroup$ @user0193: The fact that the gamblers are analyzing each other's strategies is what makes the situation a game (where payoffs are dependent on others' decisions) as opposed to a decision problem (where payoffs depend on a gambler's own decision and not on the decisions of others). $\endgroup$
    – Herr K.
    Commented Jan 2, 2022 at 15:25
  • $\begingroup$ @user0193: "Work load" or mental capacity is usually not an issue in game theory since players are assume to be perfectly rational, unless you want to specifically model the cognitive load. But this will turn the model into a behavioral game theoretic one. $\endgroup$
    – Herr K.
    Commented Jan 2, 2022 at 15:27
  • $\begingroup$ awsome! it makes perfect sense now! $\endgroup$
    – user0193
    Commented Jan 2, 2022 at 17:24

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