Best Response To 1d4

[Bertoldo Beyer]

Responsecorrespondences for all 2x2 normal form games can be drawn with a line for each player in a unit square strategy space. Figures 1 to 3 graphs the best response correspondences for the stag hunt game. The dotted line in Figure 1 shows the optimal probability that player Y plays 'Stag' (in the y-axis), as a function of the probability that player X plays Stag (shown in the x-axis). In Figure 2 the dotted line shows the optimal probability that player X plays 'Stag' (shown in the x-axis), as a function of the probability that player Y plays Stag (shown in the y-axis). Note that Figure 2 plots the independent and response variables in the opposite axes to those normally used, so that it may be superimposed onto the previous graph, to show the Nash equilibria at the points where the two player's best responses agree in Figure 3.

There are three distinctive reaction correspondence shapes, one for each of the three types of symmetric 2x2 games: coordination games, discoordination games and games with dominated strategies(the trivial fourth case in which payoffs are always equal for both moves is not really a game theoretical problem). Any payoff symmetric 2x2 game will take one of these three forms.

Games in which players score highest when both players choose the same strategy, such as the stag hunt and battle of the sexes, are called coordination games. These games have reaction correspondences of the same shape as Figure 3, where there is one Nash equilibrium in the bottom left corner, another in the top right, and a mixing Nash somewhere along the diagonal between the other two.

Games such as the game of chicken and hawk-dove game in which players score highest when they choose opposite strategies, i.e., discoordinate, are called anti-coordination games. They have reaction correspondences (Figure 4) that cross in the opposite direction to coordination games, with three Nash equilibria, one in each of the top left and bottom right corners, where one player chooses one strategy, the other player chooses the opposite strategy. The third Nash equilibrium is a mixed strategy which lies along the diagonal from the bottom left to top right corners. If the players do not know which one of them is which, then the mixed Nash is an evolutionarily stable strategy (ESS), as play is confined to the bottom left to top right diagonal line. Otherwise an uncorrelated asymmetry is said to exist, and the corner Nash equilibria are ESSes.

Games with dominated strategies have reaction correspondences which only cross at one point, which will be in either the bottom left, or top right corner in payoff symmetric 2x2 games. For instance, in the single-play prisoner's dilemma, the "Cooperate" move is not optimal for any probability of opponent Cooperation. Figure 5 shows the reaction correspondence for such a game, where the dimensions are "Probability play Cooperate", the Nash equilibrium is in the lower left corner where neither player plays Cooperate. If the dimensions were defined as "Probability play Defect", then both players best response curves would be 1 for all opponent strategy probabilities and the reaction correspondences would cross (and form a Nash equilibrium) at the top right corner.

A wider range of reaction correspondences shapes is possible in 2x2 games with payoff asymmetries. For each player there are five possible best response shapes, shown in Figure 6. From left to right these are: dominated strategy (always play 2), dominated strategy (always play 1), rising (play strategy 2 if probability that the other player plays 2 is above threshold), falling (play strategy 1 if probability that the other player plays 2 is above threshold), and indifferent (both strategies play equally well under all conditions).

While there are only four possible types of payoff symmetric 2x2 games (of which one is trivial), the five different best response curves per player allow for a larger number of payoff asymmetric game types. Many of these are not truly different from each other. The dimensions may be redefined (exchange names of strategies 1 and 2) to produce symmetrical games which are logically identical.

In evolutionary game theory, best response dynamics represents a class of strategy updating rules, where players strategies in the next round are determined by their best responses to some subset of the population. Some examples include:

Importantly, in these models players only choose the best response on the next round that would give them the highest payoff on the next round. Players do not consider the effect that choosing a strategy on the next round would have on future play in the game. This constraint results in the dynamical rule often being called myopic best response.

Instead of best response correspondences, some models use smoothed best response functions. These functions are similar to the best response correspondence, except that the function does not "jump" from one pure strategy to another. The difference is illustrated in Figure 8, where black represents the best response correspondence and the other colors each represent different smoothed best response functions. In standard best response correspondences, even the slightest benefit to one action will result in the individual playing that action with probability 1. In smoothed best response as the difference between two actions decreases the individual's play approaches 50:50.

where E ( x ) \displaystyle E(x) represents the expected payoff of action x \displaystyle x , and γ \displaystyle \gamma is a parameter that determines the degree to which the function deviates from the true best response (a larger γ \displaystyle \gamma implies that the player is more likely to make 'mistakes').

There are several advantages to using smoothed best response, both theoretical and empirical. First, it is consistent with psychological experiments; when individuals are roughly indifferent between two actions they appear to choose more or less at random. Second, the play of individuals is uniquely determined in all cases, since it is a correspondence that is also a function. Finally, using smoothed best response with some learning rules (as in Fictitious play) can result in players learning to play mixed strategy Nash equilibria (Fudenberg & Levine 1998).

3 Nc3 is best. The opening is fine for both players BUT they have to know the main ideas. 3...Qa5 is most common followed by 4. Bd2. 3...Qd6 has been getting more popular but White should just develop normally. I have played both sides of this opening many times

in fact, that kind of defense for blacks is a kind of risky,its like gamble for blacks, if white doesnt know that defense,they can surprise, but if know it, the black Queen will be jumping around a lot of time losing time, and the best response Qxd5 is Nc3, then you have to open diagonals for yours bishops and go on.....

After 1 e4 d5 2 exd5 Qxd5 3 Nc3 Qa5 4 d4 Nf6 5 Nf3 the move 5...Bf5 is what is used in the repertoire book The Scandinavian for Club Players, so 5...Bf5 is what I play since I use that book. 5...Bg4 looks playable but 5...Bf5 scores better in databases.

hi. new player here.

just curious what BLACK's best responses are to WHITE opening with pawn either C4 or F4?

I'm just not used to these opening as much and my counters don't seem very good. I often find myself in awkward positions.

I think just about every seriously chess player is less prepared for both 1. c4 and 1. f4! 1.c4 is a much more serious try than 1. f4. Against 1. f4, I would simply recommend 1. ...d5 followed by 2. ...g6, 3. ...Bg7, 4. ...c5, and 5. ...Nc6 with an equal game. Against 1. c4 white can adopt many setups, however, it is useful to find out if what you play against 1. d4 has some possible similiarities to a possible 1. c4 continuation as they may transpose. (i.e. A Caro-Kann and Slav player may choose to play 1. ...c6 against the English). Hope this helps!

First of all 1. c4 is a very serious opening allowing white to really play for the advantage. The many possible transpositions to other openings make it one of the hardest openings to master. Personally I play e5 and the revered sicilian against the english (e5 / Nf6 / d5 / Nxd5).. And although it might sound like the sicilian it is not even close. It is a fairly easy system to learn and not that easy to combat as white.

I always play 1. c4 and usually 2. NC3, it's a very interesting opening and many players aren't prepared well for it. If my opponent play 1.c4 , I'll answer 1.. c6 , because I like Slav defense and Caro-Kahn. And I don't like then answer is 1..c5, in this case it's hard to get advantage for me.

what he said was more along the lines of how amazed he was at the number of people who play 1. d4 to avoid playing the white side of the sicilian who are more than happy to reply to 1. c4 with e4 and get the same position down a tempo! iirc of course.

"It has often struck me as strange that so many players are happy to reply 1.c4 with 1..e5 and yet are unwilling to play 1.e4 themselves. It appears very odd that they are happy to play the black side of the Reversed Sicilian and not the white side of a normal Sicilian, a whole tempo up."

Though there might be some overlap between the sicilian and the reversed sicilian the difference of one tempo matters a lot. (Please note that I only play the reversed sicilian versus g3 setups and that is 80% of the time)

I play the Nimzo Indian and love it but i do avoid Nf6/e6 lines against the c4 as i do not really trust black after 1. c4 Nf6 2. Nc3 e6 3. e4 (the Minekas variation). It is a really dangerous system and even with good play for black I rather play white.

Vs. 1.f4 I like 1...c5 since it's recommended by De Firmian. Against 1.c4 I sometimes set up a Queen's Indian setup or play 1...e5 because of the potential for imbalances. The reversed dragon is tough from either end of the board however.

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