Timeline for How to find the equilibrium of the following stochastic game (numerically)
Current License: CC BY-SA 3.0
15 events
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| Sep 9, 2016 at 15:04 | comment | added | KevinKim | @MichaelGreinecker Ok, then if I allow the strategy to depend on both the label pair for the current matching pair and also the system state, i.e., the fraction of H labels within each group (e.g., a central planner always announce these 2 fractions at the beginning of each period), then how to compute at least numerically the equilibrium strategy? | |
| Sep 9, 2016 at 14:35 | comment | added | Michael Greinecker | @FTXX The decision problem will be stationary with the states being distributions, but you do not allow your startegies to depend on these solutions. Whether action 1 or 0 is optimal will in general not just depend on the types of the pair selected but the distribution. | |
| Sep 9, 2016 at 13:53 | comment | added | KevinKim | @MichaelGreinecker why it is not an MDP? let's say player only observe the (H,L) pair, i.e., using my original strategy, then when a "deal" occurs, the focal player's label will change according to P matrix, and the label of the other party he can meet in the next period also changes right? and that probability depends on the distribution of H labels in the other group, which is a consequence of all players' strategy. I agree that my strategy is not stationary in the usual sense, but the decision problem is still MDP (what I confused myself is that I have system state and individual state) | |
| Sep 9, 2016 at 13:47 | comment | added | Michael Greinecker | No, but "stationary strategies" usually mean strategies that are functions of the state. Your class of strategies does not have the property that when everyone uses such strategies, each players decision problem is a Markov decision problem. So what you do is different from what people usually do in stochastic games. | |
| Sep 9, 2016 at 13:40 | comment | added | KevinKim | @MichaelGreinecker I am very new to stochastic games, I am not sure I understand you. So what do you mean by the states make the game stationary? So in my case, as you said, the system state is the distribution of H label within each group (i.e., 2 numbers between 0 and 1), are you saying the "stationary strategy" will make these 2 distributions (i.e., 2 numbers) become a constant? e.g., under the "stationary strategy", the H label fraction in group A is always 0.4, and the H label fraction in group B is always 0.8, sth. like that? | |
| Sep 9, 2016 at 13:16 | comment | added | Michael Greinecker | @FTXX It is a standard result that you can restrict yourself to stationary strategies in a stochastic games, where stationary strategies are strategies that depend only on states and states make the game stationary. But the states in your game are the whole distribution of types and your strategies do not depend on this distribution. If everyone ele uses strategies of the kind you specified. the decison problem of a player will in general not be stationary. | |
| Sep 9, 2016 at 13:01 | history | edited | KevinKim | CC BY-SA 3.0 |
added 86 characters in body
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| Sep 9, 2016 at 13:00 | comment | added | KevinKim | @MichaelGreinecker I thought the strategy I used is pure and stationary strategy. Are you suggesting that there could be mixed and history dependent strategy? Sorry, the strategy I said in the OP is an example. I really want to know how to compute the equilibrium strategy, which could be anything. I will change my OP | |
| Sep 9, 2016 at 12:51 | comment | added | Michael Greinecker | @FTXX You make a restriction of allowable strategies and I do not see how this restriction is justified. | |
| Sep 9, 2016 at 12:35 | comment | added | KevinKim | @Michael The player only observe the label pair in his/her matching pair. I think the distribution of H within each group evolves in a deterministic way when N goes to infinity. So with the initial condition, the player should be able to derive the evolution of the distribution of H within each group at any time for any fixed strategy | |
| Sep 9, 2016 at 4:44 | comment | added | Michael Greinecker | What do players observe? And shouldn't strategies depend also at leat on the distribution of Hs and Ls? | |
| Sep 8, 2016 at 18:35 | comment | added | KevinKim | @Pcw. which means any focal player will have different probability of encountering the other group of player with label H from period to period. Hence, when he compute the expected total discounted reward before the game starts so as to figure out the best strategy, the expectation is very complicated, since it changes over time and depends on the strategy of both group of people. So how the change of the focal player's label only depends on the transition matrix, but the probability of the other party he will meet in the future depends on the collective action | |
| Sep 8, 2016 at 18:33 | comment | added | KevinKim | @Pcw. The idea is the following: think about in period 0, there are 50% of group A player with label H and 30% of group B player with label H. Then let's say group A player always choose 1 and group B player only choose 1 when his label is H and the counterpart's label is H in his pair and if his label is L, he always choose 0. Then you know that only gonna be less than 100% of pairs reach a "deal", then according to the transition matrix, the distribution of group A (and B) player with label H will change in period 1 | |
| Sep 8, 2016 at 18:07 | comment | added | Pcw. | Since the players are small (atomistic), isn't it simply the solution of a static game (played over and over)? Moreover, they never meet again, which makes it looks even more as a static game. I am probably missing something. | |
| Sep 8, 2016 at 17:18 | history | asked | KevinKim | CC BY-SA 3.0 |