Given a weight vector $w\in [0,1]^d$ such that $\sum w\_i=1$, the game goes as follows:


Two players, $X,Y$ choose strategies $x,y\in [0,1]^d$ such that $\sum x\_i = \sum y\_i = 1$.


The utility (profit) for player $X$ is given by $$u\_X=\sum\_{i:x\_i>y\_i} x\_i\cdot w\_i$$


That is summing over all coordinates in which $x$ is larger than $y$, and the profit for the coordinate is given by $x\_i\cdot w\_i$.


Y's utility is symmetric ($u\_Y=\sum\_{i:y\_i>x\_i} y\_i\cdot w\_i$).


This also interests me for low dimension (say $d=3$) if it makes it easier.




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We assume w.l.o.g that $i>j\implies w\_i\geq w\_j$.


If $w\_1\geq \frac{1}{2}$ then this is trivial, so assume this is not the case.


If needed we may assume that if $x\_i=y\_i$ then they both get $\frac{x\_i\cdot w\_i}{2}$ utility for that coordinate.




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> 
> Is there a known name for this game?
> 
> 
> Is the equilibrium of this game computable or is it $PPAD$-hard?
> 
> 
> 



What is the (mixed strategies) equilibrium of this game?
If you change the payoff to 


$$\sum\_{i:x\_i>y\_i} w\_i $$


then this type of game is called a [Colonel Blotto game or a Blotto game](http://en.wikipedia.org/wiki/Blotto_games). It was studied by Borel. Your assumption that $\sum x\_i = \sum y\_i$ is not always a requirement.


In [a 1950 paper](http://www.rand.org/content/dam/rand/pubs/research_memoranda/2006/RM408.pdf), Gross and Wagner analyzed a few variants and contributed the name. They covered the cases of 


* $d=2$ in your notation, $n=2$ in theirs.
* $d=3$, $\sum x\_i = \sum y\_i.$
* $\sum x\_i = \sum y\_i$ and $w\_j=1/d.$


Blotto games have been identified and analyzed in a few other papers.


