In the simplest asymmetric Colonel Blotto game with 2 players, dividing their given Ni soldiers (i=1,2) over 2 battlefields, what are their expected utilities, Ui (i.e., expected number of battlefield victories), in a Nash Equilibrium?


If the continuum Ni case is significantly easier to solve, you can consider the individual soldiers to be divisible. I’d also like to see an explicit Nash Equilibrium strategy for the two players, but it seems that there might be many cases, so it can’t be written down simply.



Non-Constant-Sum Blotto Game for Only 2 Players and 2 Battlefields
[O. Gross and R. Wagner, 1950. "A Continuous Colonel Blotto Game," *Rand Research Memorandum* RM-408](http://www.rand.org/content/dam/rand/pubs/research_memoranda/2006/RM408.pdf) covers this case (pages 2-4) and much more.


If the players have equal resources, then it's always a tie regardless of strategy (since you didn't assign different values to the different battlefields). 


If one player has more than twice the resources of the other, splitting them evenly guarantees a complete victory.


Suppose the players have $B$ and $E$ resources, and without loss of generality assume $2E \gt B \gt E$.


Let $m = \lfloor B/(B-E)\rfloor \ge 2$. 


The value of the game with optimal play is $2/m$. This is the difference between wins and losses, and twice the probability of a win in both battlefields, $1/m$.


Let $d= B-E, B = md + r$. Let $p$ be any number in $(r, d)$. Let $q = E/(m-1)$.


An optimal strategy for Blotto, the player with $B$, is to have a $1/m$ chance to allocate each of $p, p + d, p + 2d, ... p+(m-1)d$ in the first battlefield. 


An optimal strategy for the enemy with $E$ is to have a $1/m$ chance to allocate each of $0, q, 2q, ... (m-1)q$ in the first battlefield.


For example, if $E=1$ and $B=1.6$, then $m=2$ and the enemy can do no better than to commit everything to a random battlefield. Colonel Blotto tries to guess which one, and succeeds with probability $1/2$ (or else they split the battlefields). If $E=1$ and $B=1.4$, then $m=3$ and the enemy should allocate its forces as $(0,1)$, $(1/2,1/2)$, or $(1,0)$ with equal probabilities. Colonel Blotto can only win both battlefields in one of these situations, picks one, and guesses correctly with probability $1/3$.


