# Team learning as a game

## Abstract

A machine *FIN*-learning machine *M* receives successive values of the function *f* it is learning; at some point *M* outputs *conjecture* which should be a correct index of *f*. When *n* machines simultaneously learn the same function *f* and at least *k* of these machines outut correct indices of *f*, we have *team learning* [*k,n*]*FIN*. Papers [DKV92, DK96] show that sometimes a team or a robabilistic learner can simulate another one, if its probability *p* (or team success ratio *k/n*) is close enough. On the other hand, there are critical ratios which mae simulation o *FIN*(*p*_{2}) by *FIN*(*p*_{1}) imossible whenever *p*_{2} _< *r* < *p*_{1} or some critical ratio *r*. Accordingly to [DKV92] the critical ratio closest to 1/2 rom the let is 24/49; [DK96] rovides other unusual constants. These results are comlicated and rovide a ull icture o only or *FIN*- learners with success ratio above 12/25.

We generalize [*k, n*]*FIN* teams to *asymmetric teams* [AFS97]. We introduce a two player game on two 0-1 matrices defining two asymmetric teams. The result of the game reflects the comparative power of these asymmetric teams. Hereby we show that the problem to determine whether [*k*_{1}]*FIN* ⊂ [*k*_{2}, *n*_{2}]*FIN* is algorithmically solvable. We also show that the set of all critical ratios is well-ordered. Simulating asymmetric teams with probabilistic machines from [AFS97] provides some insight about the unusual constants like 24/49.

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