statistics - Formula/Algorithm for Weighting Game Outcomes -

i have interesting conceptual problem, , i'm wondering if can me quantify it. basically, i'm playing set of games... , each game know probability win, probability tie, , probability lose (each game have different probabilities).

at high level, want know is: games should focus attention on? example, i'm not going put effort games have 0% chance of winning (or games have 100% chance of winning). 50/50 game, care lot , want put in effort. if ties not involved, simple as: "care-ability" = how close chance of winning 50%? ties, complicates things.

i'm not sure it's strictly necessary, if need to, can assume win 0 points, tie give 1 point, , win give 2 points. in other words, valuable go loss tie, go tie win.

you can assume games independent. basically, i'm looking quantitative metric "care-ability" (a value 0 1 example).

anybody have ideas how approach this? if you're economics person, can imagine have finite number of dollars can spend on improving chances of winning games. how allocate dollars across games in order maximize expected outcomes?

thanks in advance!

edit: sorry, i've since realized poorly phrased question. don't specify relationship between additional investment , produced outcome. wanted assume linear relationship, in case, doesn't matter game invest in, since increase expected value same way. actual problem little more complicated, , need rethink bit. helped , gave great ideas!

you can formulate constrained optimization problem.

i'm going ignore draws now...

so need first let a_i amount spend on game i.

the chance of winning game presumably function of a_i .. call p_i(a_i)

your expected payout game 2 * p_i(a_i)

so total expected payout p = 2* sum( p_i(a_i) )

you have constraint on amount spend... sum(a_i) = a

your aim maximise p subject constraint.

using lagrange method gives n+1 equations solve simultaneously, unknowns a_i , lambda.

n equations this:

 2 p_i'(a_i) = lambda   

and 1 constraint equation

 sum(a_i) = total 

how solve these going depend on structure of p_i functions. depending on structure or p_i functions may need introduce aditional constraint each a_i > 0. i'd try structure p_i's avoid makes solving equations harder.

if wanted introdue chance of draw you'd split p_i(a_i) w_i(a_i) , d_i(a_i) , change payout per game 2 * w_i(a_i) + 1 * d_i(a_i) .. though doesn't change of core maths.