We have seen that Nash equilibria are likely outcomes of games. But what happens in games without Nash equilibrium in pure strategies? Even very simple games like ROCK-SCISSORS-PAPER do not have any such Nash equilibrium. Whereas in games with pure Nash equilibria players have even an interest of communicating their strategy to the other player before the game, in ROCK-SCISSORS-PAPER it is crucial to leave your opponent in the dark about what you are planning. You want to surprise your opponent, and such a surprise is best achieved by surprising yourself. This could be done by delegating the decision about your strategy to a random device. This is essentially the idea of a mixed strategy.
An example for a mixed strategy in ROCK-SCISSORS-PAPER is to play "rock", "scissors", or "paper" with probabilities 50%, 25%, or 25%, respectively. Before the game is played, the player decides randomly, based on these probabilities which pure strategy to use. Pure strategies can also be viewed as mixed strategies where one strategy is chosen with probability 100% and the others with probabilities 0%.
If we add the above mentioned mixed strategy (50% rock, 25% scissors, 25% paper) as an option for the first player in ROCK-SCISSORS-PAPER, then the expected payoffs for player 1 against player 2's pure strategies Rock, Scissors, Paper, are 0, 0.25, -0.25 respectively.
If the first player, Ann, plays the mixed strategy and Beth plays Rock, then with 50% probability there will be a tie Rock versus Rock, with 25% probability Beth will win (Beth's Rock against Ann's Scissors), and with 25% probability Ann will show Paper and win against Beth's Rock. Thus the expected payoff for Ann when playing this mixed strategy against Beth's Rock equals 50%·0 + 25%·(-1) + 25%·1 = 0. Thus the values in the fourth row are expected values of the corresponding values in the other rows and same column, using the probabilities of the mix. For instance, the second value in the fourth row is 50%·1 + 25%·0 + 25%·(-1) = 0.25, and the third one 50%·(-1) + 25%·1 + 25%·0 = -0.25. Even though this new mixed strategy doesn't dominate any of the pure strategies, the newly added mixed strategy may be attractive to a player aiming at the maximin strategy since it guarantees a payoff of -0.25 compared to -1 in the other three cases.
Of course, the other player Beth is also entitled to mixed strategies. We assume that Beth chooses a mix of 25% rock, 50% scissors, 25% paper. Inserting this 25-50-25 mix as another one of Beth's options, we obtain the following bimatrix:
The new values are computed as before, as expected values, using the payoffs of the same row, weighted by the probability of the mix. For instance, the last entry in the first row is computed as 25%·0 + 50%·1 + 25%·(-1) = 0.25. Even the payoff for 50-25-25 mix against 25-50-25 mix is computed this way, using the fourth row values, as 25%·0 + 50%·0.25 + 25%·(-0.25) = 0.0625. Maybe not too surprisingly, Ann's mix, with its emphasis on rock, beats the Beth's mix, that is heavy on scissors.
As can be seen in the matrix above, "Paper" is better against Ann's 50-25-25 mix than "Rock" or "Scissors", since it gives an expected payoff of 0.25 for Beth. The opponent emphasizes "Rock", so you must emphasize the move beating "Rock", which is "Paper". Of course we could also look for best responses among the mixed strategies, but why would we mix strategies of different value---we would just dilute the best response.
This is also true in general. Since the payoffs of a mixed strategies are weighted averages of the payoffs for the pure strategies, the maximum payoff is always achieved at a pure strategy.
The mixed strategies that are best responses are just all combinations of the best response pure strategies. To formulate this, the notion of support may be useful---the support of a mixed strategy are all pure strategies that occur with nonzero probability.
So do we even need these fancy mixed strategies, if we can always react with old-fashioned pure strategies? This is discussed next.
Assume a game is repeated over and over. How would we find out what mixed strategy the other player is using? He or she may not tell us. Well, you may want to observe very closely what the other player is doing, and react accordingly. You may want to always play the best response pure strategy to the observed mixed strategy played by the other player.
However, the other player may observe you as well, observe your strategy as a mixed strategy, and react accordingly. That means, you will adapt your play according to the play of the other player, and the other player adapts his or her play according to your observed play. Even though both players think they are always playing a pure strategy, it looks as if they would play mixed strategies, since their pure strategies are changing over time. In their histories, they play the different pure strategies with some relative frequencies, which are perceived as probabilities by the other player. These relative frequencies do change over time, but maybe eventually less and less---they might converge. This idea has been formulated by G.W. Brown [Brown 51]. He proposed that for two-person zero-sum games such a process will eventually converge and that these resulting mixed strategies are both best responses to each other.
This process can be simulated in the following Excel sheet. In ROCK-SCISSORS-PAPER, the resulting mixes of the histories are 1/3 Rock, 1/3 Scissors, and 1/3 Paper.
The following example illustrates that the pure strategies do not always occur with equal frequency:
Brown's fictitious play is important even if you play the same opponent only once. In a sense, both players would simulate thousands of rounds in their heads, and arrive at the mixed strategies. That's where the word "fictitious" comes from. Even though, after this one game, nobody can observe your probabilities by which you have chosen the pure strategies, it is important to have them and to select your move accordingly.
Up to now, the maximin strategy was supposed to be a pure strategy. From now on we also allow mixed strategies.
First note that for every mixed strategy of Ann, Ann's worst outcome is always achieved at a pure strategy of Beth. The reasoning is the same as for the fact mentioned above that among the best responses, there is always a pure strategy one. Ann's payoff for any mixed strategy of Beth is just a weighted average of Ann's payoffs for the different pure strategies of Beth, and such a weighted average can not be less than the smallest value.
For every one of the infinitely many mixed strategies of Ann, we next create the row of the payoffs versus the finitely many pure strategies of Beth. The mixed maximin strategy is the mixed strategy with the highest lowest entry in the row---it is the mixed strategy that guarantees the highest expected payoff, which we may call the mixed security level for Ann.
Take ROCK-SCISSORS-PAPER2 as example, where whenever Ann plays scissors the bid is doubled. The Payoff matrix of this zero-sum game is as follows:
We give a few rows. The formula for calculating the entries in a row is given in the last row.
|p1/p2/p3-mix||p1·0 + p2·(-2) + p3·1||p1·1 + p2·0 + p3·(-1)||p1·(-1) + p2·2 + p3·0|
Let's now move to a general 3 × 3 bimatrix.
|Up||A1,1, B1,1||A1,2, B1,2||A1,3, B1,3|
|Middle||A2,1, B2,1||A2,2, B2,2||A2,3, B2,3|
|Down||A3,1, B3,1||A3,2, B3,2||A3,3, B3,3|
This is a so-called linear program.
Using methods from this rather sophisticated topic, (which will not be covered in this book) one finds that in the example ROCK-SCISSORS-PAPER2 above, p1=2/5, p2=1/5, p3=2/5 is indeed the only mixed maximin strategy, with mixed security level of 0. For Beth, the mixed maximin strategy is q1=1/3, q2=1/3, q3=1/3.
Let's go back to maximin value and strategy. Ann will play her maximin strategy if she assumes that all other players would always find moves that would minimize Ann's payoff for her move chosen. Of course, such fear is a little paranoid for two reasons: First, why would the opponents know what move Ann choses, after all the game is simultaneous, and secondly, why would they even bother, usually they want to maximize their payoff, not minimize Ann's (although in two-person zero-sum games this is the same). Now if if the other players really would want to punish Ann, they would select moves such that, no matter what Ann choses, she still can get only limited payoff. For each move combination of all players except Ann, we look at all payoffs for Ann (!) for Ann's different options. We choose the maximum of these payoffs for each such move combination. Then we select that move combination with the minimum such value. This is called the punishment value for Ann.
Remember that in a two-player game, the maximin value for Ann is obtained by looking on Ann's payoffs in the rows, selecting the lowest such value in each row, and then selecting the row with the maximum such minimum. In order to obtain the punishment value for Ann, we also look only on Ann's payoffs, but we start with the columns. In each such column we fix the maximum payoff in each column, and after that we select the column with the minimum such number. The role of rows and columns is reversed, and the ordering of Min and Max is reversed; for this reason the punishment value is sometimes also called the minimax value.
If we allow all mixed strategies, we even get equality in case of only two players:
However, this is not true for three or more players.
For two-person zero-sum games, there follows the somewhat strange fact that in some situations, if your opponent is supposed to mix all pure strategies, then it doesn't even matter whether she mixes, or which pure strategy she uses, as long as you stick to your mixed strategy. This happens in ROCK-SCISSORS-PAPER: If you use all three possibilities with probability 1/3, then anything your opponent does gives an expected outcome of 1/2 for each. However, in more complex situations still "most" mixed strategies are inferior against a mixed strategy that is part of a Nash equilibrium. Examples are VNM POKER(4,4,3,5), for instance, and also Morra.
A Nash equilibrium of mixed strategies is called a mixed Nash equilibrium.
These special games have two nice features:
Actually von Neumann's proved a more general result, his famous "Minimax Theorem", from which the theorem above follows easily. Such a Nash equilibrium can also be obtained by Brown's fictitious play process described in the subsection above:
Things become more complicated in these cases. Let me illustrate this with the following two examples, one game is zero-sum, the other isn't:
Therefore this strategy is the maximin strategy for Ann if mixed strategies are allowed. But since the maximin strategy and value only depend on Ann's payoffs, and these don't change between Game 1 and Game 2, the strategy with 1/3 "Up" and 2/3 "Down" is also the maximin strategy for Ann in Game 2 (with mixed strategies allowed). Since Game 2 is symmetric, Beth has a similar maximin strategy of choosing 1/3 "Left" and 2/3 "Right". The expected payoff for both when both play this strategy is 10/3. But this is not a Nash equilibrium in Game 2---every deviation of Ann in the direction of more "Up" is rewarded, as is every deviation of Beth in the direction of more "Left". Actually we will show below that Game 2, if mixed strategies are allowed, has three mixed Nash equilibria:
Although the maximin strategies do not necessarily form a mixed Nash equilibrium for general games, mixed Nash equilibriua always exist, even for non-zero-sum games with an arbitrary number of players:
The mathematician John Forbes Nash Jr. (*1928) obtained this result in his Ph.D. dissertation He was awarded the 1994 Nobel Prize in Economics (shared with Reinhard Selten and John Harsanyi) for this result and some other papers written around 1950-1954. Mathematically more challenging were his results about questions in differential geometry (Every Riemann manifold can be embedded into some Rn) and the theory of partial differential equations (1957: Theorem of de Giorgi and Nash, answering one of Hilbert's problems). In the sixties he was candidate for the Fields medal, the Mathematics equivalent of the Nobel Prize. According to John Milnor, "Nash's prize work is an ingenious but not surprising application of well-known methods, while his subsequent mathematical work is far more rich and important" [Milnor 1998]. Nash was suffering from paranoid schizophrenia for a long time.
von Neumann's achievement in Game Theory was to put the topic on the agenda and of course, his famous Minimax Theorem. However, von Neumann and Morgenstern's monograph concentrate very much on two-person zero-sum games, which really don't occur that often outside of literal games. Nash's achievement in Game Theory was to clarify the distinction between cooperative and non-cooperative games, to shift emphasis from two-person zero-sum games to general non-cooperative games, and to show that every such game has a (Nash-) equilibrium in mixed strategies. Furthermore, he introduced four axioms for bargaining that guarantee a unique solution.
It is relatively easy to check whether a bunch of mixed strategies, one for every player, forms a Nash equilibrium: According to the Indifference Theorem, all we have to do is to find for every player all pure strategy best responses to the mixed strategies of the others, and check whether all pure strategies in the support of that player belong to them. But the question is how to find them. And this is in general not easy, even when the normal form is given. Without that, if the game is described by its extensive form, even calculating the normal form may be not feasible, since it may be far too large. An example of such a game is DMA Soccer II. In the following we assume that the normal form is given.
What should be done first is to eliminate all strictly dominated pure strategies from the normal form:
As mentioned above, for the special two-player zero-sum games, the problem to find Nash equilibria in mixed strategies can be formulated as a Linear Program. And for such problems, George Dantzig developed around 1947 a solution method, the so-called "Simplex Algorithm." This method is in practice quite fast, although there are a few artificial cases where the algorithms needs long. But later, other algorithms have been found that always find a solution in time bounded by a polynomial in the number of rows and columns.
For general games, if the numbers of rows and columns are small, the Indifference Theorem can be used. Then it suffices to solve a few systems of linear equations to obtain all mixed Nash equilibria. We will describe details below for two-player games and bimatrices of sizes 2 × 2, 2 × n, 3 × 3, and also for three players with two options each.
This method gets infeasible for large number of options. Although in a 2 player game where the normal form has 10 rows and columns, we need to solve systems with up to 10 equations and 10 variables, which is not too difficult using technology, the number of such systems we need to solve raises already to more than 1,000,000. More sophisticated approaches are needed. For 2-player games, the best approach so far is an algorithm found by Lemke and Howson. By the way, they also showed that under that conditions the number of Nash equilibria must be odd ---except in degenerate cases. Still, this algorithm has exponential running time in the worst case [Lemke/Howson 1964].
Finally Brown's fictitious play can be used to get approximate values of mixed Nash equilibria. However, the method does not always work. Moreover convergence may be slow---even after say 50000 iteration you may not be within say 1% of the probabilities. And if there are more than one mixed Nash equilibria, it may be that Brown's process converges only to one of them in most cases.
Here is an applet where you can compute mixed Nash equilibria in zero-sum games. For non-zero-sum games you can use this one.
In the following we will discuss methods for several small cases.
If Ann mixes between two choices, the probability p for her first move obviously implies that the probability for her second move must be 1-p. For such a mix, Ann's payoff provided Beth plays some of her pure strategies is a linear function in p, with the cases p=0 and p=1 being the payoffs in the pure cases. So for Beth's ith move we draw the straight line from (0,A1,i) to (1,A2,i). We then draw the curve always following the lowest of these lines. The largest value of this curve is the security level for Ann, and the corresponding p-value belongs to Ann's Nash equilibrium mix.
Look at the following example:
The Nash equilibrium p is about 0.55, and Ann's guaranteed payoff for this mix is a little more than 1.5. Since the resulting point lies at the intersection point of the straight lines belonging to B1 and B3, Beth will mix between these two pure strategies.
A similar approach, but now in 3 dimensions, works for the 3 × n case. Ann's mix would be p1, p2, p3 with p1+p2+p3=1. Any such triple of numbers corresponds to a point on a triangle, using barycentric coordinates. These are calculated as the ratio of shortest distance of the point to one side over the shortest distance of the third point of the triangle to that same side. Now we build straight lines perpendicular to the triangle at the three points, and hang a triangle at these pillars at heights A1,i, A2,i, A3,i for every pure strategy i of Beth. These triangles are intersecting. The point on the triangle having most "air" above it until it meets the lowest ceiling is Ann's maximin mix, and the height until this lowest ceiling there is Ann's security level for mixed strategies.
In the example to the right the triangle and the three pillars are drawn in white. The matrix is
The simplest normal forms are for two-players games where each player has exactly two strategies. That is, we are facing a payoff matrix of the form
|Up||A1,1, B1,1||A1,2, B1,2|
|Down||A2,1, B2,1||A2,2, B2,2|
There are three cases of mixed Nash equilibria: One where Ann mixes and Beth doesn't, one where Beth mixes and Ann doesn't, and one where both mix.
Let us look at this third case first. Assume that Ann chooses "Up" with probability p, 0 < p < 1 and "Down" with probability 1-p. In the same way, Beth chooses "Left" with probability q, 0 < q < 1 and "Right" with probability 1-q. The Indifference Theorem above implies that Ann faces the same expected payoffs when playing Up or Down provided Beth keeps mixing. These expected payoffs are q·A1,1+(1-q)·A1,2 and q·A2,1+(1-q)·A2,2. Therefore
The other two cases are easy to analyze: Assume Ann mixes, plays "Up" with probability p and "Down" with probability 1-p, but that Beth plays the pure strategy "Left". Then the Indifference Theorem implies that A1,1=A2,1. Then both "Up" versus "Left" as well as "Down" versus "Left" are pure Nash equilibria, and every value of p between 0 and 1 would produce a mixed strategy for Ann that would form a Nash equilibrium with "Left". Therefore we would have infinitely many mixed Nash equilibria, with two pure ones as extreme cases. The other cases are similar. So ordinarily we would have at most one mixed Nash equilibrium, with both Ann and Beth really mixing, or we would have infinitely many of them.
The complete analysis of the 2 × 2 case --- domination, best response, all pure and mixed Nash equilibria --- can be done with the Excel sheet Nash.xls on the sheet "Nash22".
The general payoff bimatrix in this case is:
|Up||A1,1, B1,1||A1,2, B1,2||A1,3, B1,3|
|Middle||A2,1, B2,1||A2,2, B2,2||A2,3, B2,3|
|Down||A3,1, B3,1||A3,2, B3,2||A3,3, B3,3|
We have one case where Ann mixes between all three strategies, three cases where Ann mixes just between two of them, and three cases where Ann uses a pure strategy, and the same holds for Beth. Thus we can in principle pair any of Ann's seven cases with any of Beth's seven cases to get 49 possible patterns for Nash equilibria.
Let us start with the most interesting pattern where both Ann and Beth mix between all their three strategies. Assume that Ann chooses "Up" with probability p1, "Middle" with probability p2, and "Down" with probability 1-p1-p2. In the same way, Beth chooses "Left" with probability q1, "Middle" with probability q2, and "Right" with probability 1-q1-q2. The Indifference Theorem above gives us two double equations, namely
The next case is where Ann mixes between two strategies, say between "Up" and "Middle" with probabilities p and 1-p, and Beth between all three as before. We get three equations with three variables
If both Ann and Beth mix between two strategies, essentially the formulas for the 2×2 case can be used.
In the same way as above, if one player mixes and the other plays a pure strategy in a Nash equilibrium, some of the payoff coefficients of the matrix, in a row or a column, are identical, and the mixing player could use any mix between the strategies involved to get a Nash equilibrium.
The Indifference Theorem above is also the key for finding all Nash equilibria in mixed strategies in three-players games with two pure strategies each, but we will see that and why this case is much more complicated than the previous two cases. Being more complicated than the two-players 2×2 case is probably not surprising, but why is it more complicated than the two-players 3×3 case, which has 18 parameters? Our case considered has 24 parameters. ..................
.......... ROCK-SCISSORS-PAPER has an obvious second mover advantage. The following is a another game with second mover advantage. There are no pure Nash equilibria
And here is a game, without pure Nash equilibria, where Ann has a first mover advantage and Beth a second mover advantage.