game_theory: Add lemke_howson

[oyamad]

Implement the Lemke-Howson algorithm.

1. Implemented with floating point arithmetic. I also tried integer pivoting (as described in von Stengel 2007), but overflow easily occurs with np.int64. In a Julia version we may want to try with BigInt.

2. The full_output option mimics root finding routines in scipy such as brentq.

3. sicpy.optimize.OptimizeResult is used again as in DPSolveResult (but with sort in __repr__). Maybe we should put it in util for example, to be inherited by DPSolveResult and NashResult here to get rid of duplicates in the code.

Cc: @kevtimova @arnavsood @shizejin

[arnavs]

@oyamad: Thanks for notifying me about this.

Is the Julia analog already being worked on? If not, I'd be happy to attempt it.

[jstac]

@oyamad It's beautiful code, as usual.

The failing tests seem to be spread across the library -- perhaps due to changes in Numba?

[cc7768]

I can confirm it is due to changes in numba -- I was trying to run some of the linear state space code and had issues until I downgraded to numba 0.28.

I'll raise a separate issue so we can address this.

[jstac]

@cc7768 Thanks, that would be helpful. I saw somewhere that numba expects new style classes now. Could that be the issue? (In other words, classes shouldn't inherit from object.)

[oyamad]

@arnavsood No I don't have a Julia version. You can continue your work in Julia.

[coveralls]

Coverage decreased (-2.1%) to 84.615% when pulling ebdc888 on lemke_howson into 120ab73 on master.

[coveralls]

Coverage decreased (-2.1%) to 84.615% when pulling 7867fad on lemke_howson into 120ab73 on master.

[coveralls]

Coverage decreased (-2.006%) to 83.37% when pulling 6202d16 on lemke_howson into e0bdaa7 on master.

[oyamad]

Rebased on master.

[jstac]

The code is beautifully written, everything works as advertised and the documentation is good. Thanks!

@mmcky My view is that this can be merged.

@oyamad Might you or one of your students be willing to add a notebook to the notebook site giving a couple of examples, or, alternatively, update this notebook accordingly?

[mmcky]

Thanks @oyamad and @jstac. I will merge this PR now.

[oyamad]

Sure, I will update the game theory notebook.