I'm probably missing the point, but I'm not sure about what you mean when you talk about two decisions to be made in the context of a static game.  As both decisions have to be made simultaneously, without knowing anything about the other player's decision, can't you simply map them into a single decision?  

 Dear Dr. Koboldt,

Thanks very for your answer!

Sorry that I might haven't described the problem clearly enough. When I say each player have two decisions to make, I mean one player have to make two decisions simultaneously, without knowing  other player's decisions.  

For example, there are two players, player A and player B. player A need to simultaneously decide how much time to spend on activity 1 and how much time to spend on activity 2, without knowing the strategy of player B. Similarly, player B need to simultaneously decide how much time to spend on activity 1 and how much time to spend on activity 2 as well, without knowing the strategy of player A. This is a static game, and one player has to make two decisions. 

May you please tell me more about mapping them into a single decision? I am not quite sure about this.  

Thanks a lot!

That game can have more than one Nash-Solution. See Battle of Sexes game. There are two Nash solutions. Perhaps, I dont't understand the question, excuse me!

"That game" is not really specified.  As far as the decision of how much time to spend on activity 1 and activity 2 goes - that's a single decision fully compatible with the setup in the Rosen paper where the strategy of a player is an m-dimensional vector.  There really isn't any added complication here - the dimensionality of the decision doesn't matter.

Hope that helps...

Christian

To prove the uniqueness of a Nash equilibrium in this context, you only need to show that it's the best strategy for each one player, given that they don't know the strategy the other player is going to choose. The stability of Nash equilibria depends more on that assumption of incomplete information than any other. But again, your question is not fully specified; therefore, answering it may mean that we understand something that's not said! Anyway, state the available strategies and their yields for each player and the answer may be clearer. 

With what I read my commnets are the following:

1.     The strategy is multi dimensional vector, where time on activity 1 and 2 are components. The Nash point is one of the solutions for 2-players game with contradictory interest.

2.     If activities 1 and 2 are the strategies of different players, then this is a game of four players, where two pairs of players A1- A2 and B1- B2 are playing with non-contradictory interest and another four cross pairs (A1-B1, A1- B2 and A2-B1, A2-B2) are playing the standard game with contradictory interests. For such more complicated game the Nash point also exists, but needs more refined interpretation.

I do not remember the details (I’ve been involved in the subject more than 30 years ago), but you might check in publications devoted to Theory of Games with Non Contradictory Interests. I would recommend the books by Yuriy Germayer.

With what I read my commnets are the following:

1.     The strategy is multi dimensional vector, where time on activity 1 and 2 are components. The Nash point is one of the solutions for 2-players game with contradictory interest.

2.     If activities 1 and 2 are the strategies of different players, then this is a game of four players, where two pairs of players A1- A2 and B1- B2 are playing with non-contradictory interest and another four cross pairs (A1-B1, A1- B2 and A2-B1, A2-B2) are playing the standard game with contradictory interests. For such more complicated game the Nash point also exists, but needs more refined interpretation.

I do not remember the details (I’ve been involved in the subject more than 30 years ago), but you might check in publications devoted to Theory of Games with Non Contradictory Interests. I would recommend the books by Yuriy Germayer.

Everyone, thanks so much for your kind help!!!  

I tried to specify the game from a mathematical perspective in the attached PDF document.  Because the constraints are linear and the payoff functions are strictly concave in terms of its own strategy while holding the other player's strategy fixed, I guess there might be some simple methods to prove the uniqueness of Nash equilibrium solution, but not quite sure about this. 

Dear Dr. Koboldt,

Yes, you are right! It is single decision of a two-dimensional vector. Many thanks! 

You have to use the Kakutani fixed point theorem. Standard game theory textbooks (e.g. Fudenberg and Tirole) have the formal proof.

Dear Prof. Martini,

Thanks very much for your kind help!

I would check whether the game may have a Nash Equilibrium in strictly dominant strategies. Perhaps you know one equilibrium already. If it is in strictly dominant strategies, then it is unique. Another possibility, which was mentioned already, would be to show that the best-reply correspondence has only one fixed point. If your game is such that the best-reply correspondences (which maps each vector of actions into another vector of best replies to those actions) are ordinary functions (i.e. not set-valued), then you have to identify the set of fixed-points of that function and show that it is a singleton.

Dear Prof. Biermann,

Thanks very much for your answer. It is very helpful. I have been trying to use the strictly dominant strategy method. 

I am curious that if the payoff function of each player is a continuous two-part piecewise function, then the strictly dominant strategy method might not be applicable because the payoff function is not differentiable at the section point. In this kind of situation, maybe the fixed point theorem is suitable. But I am not sure about this. 

The properties of the payoff function (i.e. whether it is a continuous, tow-part piecewise function and whether it is differentiable or not) do not play a role. The strictly dominant strategy approach holds for all payoff functions. The caveat with the strictly dominant strategies is that in many games there is no Nash equilibrium in strictly dominant strategies. If that is the case, you cannot apply this property of the Nash equilibrium to show that it is unique.

I looked at the description that you have uploaded. Do you know anything about the functions fa and fb? 

When i understand it well, the problem is uniqueness in case of higher dimensional strategy sets. Existence seems not to be a problem (by Nikaido-Isoda etc). So the problem is equilibrium semi-uniqueness, i.e. that there exists at most one equilibrium.

Here is a rerence to a result that may be helpfull:

http://www.mathnet.ru/php/contents.phtml?wshow=issue&jrnid=cgtm&year=2014&volume=7&option_lang=rus

See the general result in the article Uniqueness of Coalitional Equilibria. This result assumes differentiability. However, it should be possible to generalize it to semi-differentiability (as suggested by the references in this article).

Dear Prof. Biermann,

Thanks a lot! 

I am sorry that I might have  mistaken the strictly dominant strategy approach as Rosen's diagonally strictly dominant method. 

Yes, the function is log form, I have changed the uploaded document and specified the functions. 

Besides, can anyone share with me the solution algorithms that are applicable for this kind of game? Thanks in advance!

in your new file, can we assume that aa1, aa2, aa3 >=0 and ab1, ab2, ab3 >=0?

Dear Prof. Biermann,

Yes, we can assume that.

Without specificying the parameters, you cannot be sure that the problem has only one NE. However, I think that all NEs will have the same payoffs for both players. Moreover, you have to carefully construct an instance of this problem to get more than one NE by choosing the parameters accordingly. Hence, for concrete given parameters, which are not chosen to generate multiple NEs, it is unlikely that there is more than one NE.

This is the argument: Let's only consider Player A (the argument would be symmetrical for Player B). First consider the case M = 0. In this case, it is a pure optimization problem without strategic interaction. It may be that there are multiple solutions (xa1,xa2) which maximize the function fA. Hence in this case no uniquness.

Now assume that M is not 0. Assume (xa1,xa2)  and (xa1',xa2') are two different solutions of fA. If M

Dear Prof. Biermann,

Thanks very much! Currently, I only have to deal with the case that M is not 0. And yes, I have to choose the parameters carefully!

It is not true... There are games with many equilibria

What does mean "each player has two decisions to make"? Every individual chooses only one strategy. For existence of Nash equilibrium you need convex compact strategy set for each player, pay-off functions are continuous and cocave over induvidual strategies. Look at Pirsoner's dilemma for non- uniqueness example in pure and mixed strategies. Good luck!