Challenge donkey-lost in maths

For this part of my challenge I wanted to get to grips with the medium pairs problem when you have around 17bb and usually one limper to contend with. I looked at various scenarios where you limp and then give yourself the option of hitting a set, shoving low flops, shoving over cbets, or check folding to overcards etc. This is essentially the problem I faced in the MTT I busted out of this morning holding 99. My conclusion is that the limp play is a small losing play each time you make it and should therefore not be entertained, but it took me a long time to work this out using lots of expectation calculations. In low stakes tournaments you see a lot of medium pairs played as a 'stop and go' in these sorts of situations and I just cannot justify this move numerically. I have run out of momentum in my maths session and may have to concede that this problem remains open ended for the time being. The crux of the problem is that you have around 54% equity if you shove and get heads up against a typical range, so you know as soon as you get these hands that you are going to have to flip for your tournament life and give two high cards the chance to call and see all 5 board cards. Given the pot to stack ratio you rarely leave any player any other option but to call if you shove as you will typically offer them around 40% break-even percentage anyway. This is before you consider that if you do it from middle position then you may well be called from behind as well, dropping your equity share to around 32% and leaving the initial limper 30% break even percentage and a very inviting call. If you fold your puny middle pair you dodge a bullet for sure but one more pass through the blinds takes you down to 15.5 bb and if this happens again the pot odds you offer become even more juicy to villains with their finger poised on the all in button. It is perhaps looking more and more like you can justify the all move unless you are in a sit and go with no antes and you have more of an edge in the shove/fold war that will play out as the bubble fast approaches. In an MTT it is probably better to shove and let the chips fall where they may. This topic needs further research and I have yet to consider cases where you face a raise, but in working this out so far I have learned how to make expectation calculations properly. I have also discovered how to predict the number and range of hands villains may hold, how to calculate certain basic probabilities and I have used up most of the rough paper I had in the living room for taking telephone messages. Good times:)