Jolene’s Reviews > Two-Person Game Theory > Status Update

I 2x2 games w/o sdlpt, mixed strat is unique(p1, q1). If one plays minmax, same expected payoff regardless of other plyr's move. If 2x2 has sdlpt, at leas 1 plyr has domin. strat. If neither has domin. strat, no sdlpt. Hard to extend to lgr games. (82)

Section ends with notes on mixed prudent strategies and involvement of risk to attain lgr payoffs than 0 (exp value over long-run in 2pp ZSG).

Minmax strats are certainly prudent (pure prudent=saddlepoint if coincident)

Mixed strats in 2pp ZSG will have value of game=0, EP's to plyrs will cancel exactly for right choice of p,q.

"In strat mix of game w/ saddlept, prob=1 to strat w/ saddlept and 0 to every other." (74)
There exists best strat for ea plyr in 2pp ZSG. If saddlept, at least one (if not more) best strats for ea. No sdlpt, no best pure->mix

Excellent coverage of mixed strats. beginning with game o imperfect info (variant of Castor-Pollux) with saddlept. soln (it persists-rare case...look for this first). So perfect nfo not reqd for saddlept but is sufficient.

Next step--extend minmax to games w/o saddlepts. Look for mixed strats. In ZSG quickly recognize lack of saddlept when "min entry in neither row is maximal in its column"-2 minmax not coincident

Minmax principle form p60:If a 2pp ZSG has saddlepts, the best ea plyr can do (assume rationality)is to choose the strat (row or column) whic contains the saddlept. AND "balance of power":best that either of the players can do, given that he is playing a rational opponent (59) --[min] guaranteed payoff (58); same values also saddlepts. regardless of how you got there.

Finally understand how they count strats. thanks to the Castor-Pollux square-the-diagonal game.
Next he moves onto dominating strats."A strategy which dominates every other is unconditionally the at least as good as or possibly better [regardless of what the other player does BUT]" best does not mean dominates. We decide on "best" against a player assumed to be "rational." Rev. of solution by domin. strats, IEDS...

Have struggled with this quantification of strategies a lot! Will try to come back to this after today's reading session.

"Utility, as it is defined in game theory, is that quantity whose expected gain in a risky choice is expected to be maximized by a decision-maker [rational]". Max of expected gain is useful principle if we're averaging over many plays but in one play can't guarantee that avg. payoff. We also see problems of joint utility maximization (which requires coordination/possibly enforceable transfer agreements)

Again def of rational players (as in TSchelling):"able to choose consistently among all possible risky outcomes." Unrealistic condition but easier to assume if outcomes all in $ (same units, etc)

Useful description of payoffs as utilities on an interval scale (perhaps derived from an exercise such as entering a lottery for the wines A and D over C, etc).

Review: 1) games of perfect information-all choice of all players known to everyone as soon as they are made (chess, checkers, tic-tac-toe)-->makes it easier to discern best way to play (w/o worrying about "chance")
AR reqs at least two bona fide plyrs to (1) make choices, (2) receive payoffs
Nice example of non-game scenarios: slot machine (you make no choices) or Solitaire (house makes no choices).