I think in this case it likely trends towards the 20-20 equilibrium, since the row-player has a preference for that. If the column-player picks 20$, then obviously the row player prefers to get 4$ EV vs 0$ EV, but the gap between $9 EV and $8 EV if the column-player picks $10 is much smaller.
I guess this is a great example of how important it is to know what all the other players value. if you don't know their payoffs, or at least the probability distribution of their payoffs (like your interim dominance example of one player either being a prisoners dilemma type or a stag hunt type but the other guy at least knows the probability), it becomes difficult to come up with the optimal tactic.
There should be third equilibrium where you choose to bid 10$ with a 40% probability (and 20$ with a 60% probability), if you expect the other player to bid 10$ with an 80% probability (and bid 20$ with 20% probability), if I am not mistaken. Interesting puzzle, as always. In this case, talk is not cheap: if agents can coordinate to choose the best outcome (which happens to be the best equilibrium for them as well), why wouldn't they do it? Reality is messy.
That's right. Per the odd rule ( ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-RhSaq97YjbA.html ), there is a third equilibrium, and it is in mixed strategies.
hi who decides the actual prices of a stock ?for example if news comes out do any owners of the stock decide and can freely ask what ever they want for the stock which drives the stock price up? or is there some kind of rule or term they must follow to raise the price of a stock? who decides these things? OR is price determined by strict supply demand rules by the float/ outstanding shares available?
I would simply write a contract where $10 are held in escrow from each bidder and forfeited to the opponent if the bidder bids $20. Should ensure compliance and I get the maximum $9 EV, plus I don't have to do any math.
The mixed strategy equilibrium (row player offering 10$ with probability 40%; column player offering 10$ with probability 80%) seems reasonable, because it takes into account the 'valuations' (row player's 28$; column player's 24$). In general, if row player valued the book at a$ and column player at b$, both in [21,29], then row player would give 10$ with probability b/10-2 and column player would give 10$ with probability a/10-2. What strikes me is that the probability for each player's offer, only depends on the other player's valuation.
Yep! And that's a general property of mixed strategies: you need to choose a mixture that induces indifference from your opponent. Thus, small changes to your payoffs do not change your strategy, only changes to your opponent's. (Large changes to your payoffs can give you a dominant strategy, hence the "small" caveat.)
I don't think you can talk about "net profit" until the new owner sells the book, and then it's only worth what someone else is willing to pay for it. So if I "value" the book at $24, that's essentially meaningless. Do I want the book, or am I buying it because I think someone will show up who wants it more than I do? I'm sure this is illustrating something useful about game theory, but the story does not seem to make any game theory principle clearer to me.
