If the game does not have a saddle point, then there are difficulties in determining the price of the game and the optimal strategies of the players. Consider, for example, the game:
In this game and . Therefore, the first player can guarantee himself a payoff equal to 4, and the second can limit his loss to 5. The area between and is, as it were, a draw, and each player can try to improve his result at the expense of this area. What should be in this case optimal strategies players?
If each of the players uses the strategy marked with an asterisk ( and ), then the gain of the first player and the loss of the second will be equal to 5. This is disadvantageous for the second player, since the first wins more than he can guarantee himself. However, if the second player reveals in some way the first player's intention to use the strategy , then he can apply the strategy and reduce the first player's payoff to 4. However, if the first player reveals the second player's intention to use the strategy , then using the strategy , he will increase his payoff to 6 Thus, a situation arises when each player must keep secret the strategy that he is going to use. However, how to do it? After all, if the game is played many times and the second player applies the strategy a all the time, then the first player will soon figure out the intention of the second and, having applied the strategy, will have an additional payoff. Obviously, the second player must change the strategy in each new game, but he must do this in such a way that the first one does not guess which strategy he will use in each case.
For the random selection mechanism, the gains and losses of the players will be random variables. The result of the game in this case can be estimated average loss of the second player. Let's go back to the example. So, if the second player uses the strategy and randomly with probabilities 0.5; 0.5, then with the strategy of the first player, the average value of his loss will be:
and with the strategy of the first player
Therefore, the second player can limit his average loss to 4.5 regardless of the strategy used by the first player.
Thus, in a number of cases it turns out to be expedient not to outline a strategy in advance, but to choose one or another randomly, using some kind of random selection mechanism. strategy based on random selection, called mixed strategy, in contrast to the intended strategies, which are called pure strategies.
Let us give a more rigorous definition of pure and mixed strategies.
Let there be a game without a saddle point:
Let us denote the frequency of using the pure strategy of the first player by , (the probability of using the i-th strategy). Similarly, we denote the frequency of using the pure strategy of the second player by , (the probability of using the j-th strategy). For a game with a saddle point, there is a solution in pure strategies. For a game without a saddle point, there is a solution in mixed strategies, that is, when the choice of strategy is based on probabilities. Then
Lots of pure 1st player strategies;
Many mixed strategies of the 1st player;
Lots of pure 2nd player strategies;
Lots of 2nd player mixed strategies.
Consider an example: let there be a game
The second player chooses a probability 
. Let us estimate the average loss of the second player when applying strategies and , respectively.
Among the finite games that have practical value, games with a saddle point are relatively rare; more typical is the case when the lower and upper price - games are different. Analyzing the matrices of such games, we came to the conclusion that if each player is given a choice
one - the only strategy., then, based on a reasonably acting opponent, this choice should be determined by the minimax principle. Adhering to our maximin strategy, we certainly guarantee ourselves a payoff equal to the lower price of the game, a, for any behavior of the opponent. A natural question arises: is it possible to guarantee yourself an average payoff greater than a if you apply not just one "pure" strategy, but alternate several strategies at random?
Such combined strategies, consisting in the application of several pure strategies alternating according to a random law with a certain ratio of frequencies, are called mixed strategies in game theory.
Obviously, each pure strategy is a special case of a mixed one, in which all strategies except one are applied with zero frequencies, and this one - with a frequency of 1.
It turns out that, by applying not only pure but also mixed strategies, it is possible to obtain a solution for each finite game, i.e., a pair of such (generally mixed) strategies such that when both players apply them, the payoff will be equal to the price game, and in case of any one-sided deviation from the optimal strategy, the payoff can only change in the direction unfavorable for the deviant.
The stated statement is the content of the so-called main theorem of game theory. This theorem was first proved by von Neumann in 1928. The known proofs of the theorem are relatively complex; therefore, we present only its formulation.
Every finite game has at least one solution (perhaps in the realm of mixed strategies).
The payoff resulting from the decision is called the price of the game. It follows from the main theorem that every finite game has a price. Obviously, the value of the game v always lies between lower price game a and the top price of the game :
Indeed, there is a maximum guaranteed payoff that we can secure for ourselves using only our own pure strategies. Since mixed strategies include, as a special case, all pure ones, then, allowing, in addition to pure ones, also mixed
strategy, we, in any case, do not worsen our capabilities; hence,
Similarly, considering the capabilities of the opponent, we show that
whence the required inequality (3.1) follows.
Let us introduce a special notation for mixed strategies. If, for example, our mixed strategy consists in applying the AL strategies, with frequencies and we will denote this strategy
Similarly, the adversary's mixed strategy will be denoted by:
where are the frequencies at which strategies are mixed
Suppose that we have found a solution to the game consisting of two optimal mixed strategies S, S. In the general case, not all pure strategies available to a given player are included in his optimal mixed strategy, but only some of them. We will call the strategies included in the player's optimal mixed strategy his "useful" strategies.
It turns out that the solution to the game has another remarkable property: if one of the players adheres to his optimal mixed strategy 5 (5). then the payoff remains unchanged and equal to the price of the game v, regardless of what the other player does, if he. only does not go beyond its "useful" strategies. He, for example, can use any of his "useful" strategies in pure form, and can also mix them in any proportions.
Let's prove this statement. Let there be a solution to the game . For concreteness, we will assume that the optimal mixed strategy consists of a mixture of three
"useful" strategies respectively consists of a mixture of three "useful" strategies
and It is stated that if we stick to strategy S, then the opponent can apply strategies in any proportions, and the payoff will remain unchanged and will still be equal to the price of the game
The mixed strategy SA of player A is the application of pure strategies A1, A2, ..., Am with probabilities p1, p2, ..., pi, ..., pm and the sum of the probabilities is equal to 1: The mixed strategies of player A are written as a matrix or as a string SA = (p1, p2, ..., pi, ..., pm) Similarly, the mixed strategies of player B are denoted: , or, SB = (q1, q2, ..., qi, ..., qn ), where the sum of the strategies occurrence probabilities is equal to 1: Pure strategies can be considered as a special case of mixed strategies and can be given as a string in which 1 corresponds to a pure strategy. Based on the minimax principle, optimal solution(or solution) of the game: this is a pair of optimal strategies S*A , S*B in the general case mixed, with the following property: if one of the players adheres to his optimal strategy, then it cannot be profitable for the other to deviate from his. The payoff corresponding to the optimal solution is called the value of the game v. The price of the game satisfies the inequality: ? ? v? ? (3.5) where? And? - the lower and upper prices of the game. The following main theorem of game theory is valid - Neumann's theorem. Every finite game has at least one optimal solution, perhaps among mixed strategies. Let S*A = (p*1, p*2, ..., p*i, ..., p*m) and S*B = (q*1, q*2, ..., q* i, ..., q*n) - a pair of optimal strategies. If a pure strategy is included in the optimal mixed strategy with non-zero probability, then it is called active. The theorem on active strategies is valid: if one of the players adheres to his optimal mixed strategy, then the payoff remains unchanged and equal to the cost of the game v, if the second player does not go beyond his active strategies. This theorem is of great practical importance - it gives specific models for finding optimal strategies in the absence of a saddle point. Consider a 2×2 game, which is the simplest case of a finite game. If such a game has a saddle point, then the optimal solution is the pair of pure strategies corresponding to that point. A game in which there is no saddle point, in accordance with the main theorem of game theory, the optimal solution exists and is determined by a pair of mixed strategies S*A = (p*1, p*2) and S*B = (q*1, q*2) . In order to find them, we use the theorem on active strategies. If player A sticks to his optimal strategy S "A, then his average payoff will be equal to the price of the game v, no matter what active strategy player B uses. For a 2 × 2 game, any pure strategy of the opponent is active if there is no saddle point. Player A's payoff (loss of player B) - random variable, expected value(average) which is the price of the game. Therefore, the average payoff of player A (optimal strategy) will be equal to v for both the 1st and 2nd strategies of the opponent. Let the game be given by the payoff matrix Average payoff of player A if he uses the optimal mixed strategy, and player B - pure strategy B1 (this corresponds to the 1st column payment matrix P) is equal to the value of the game v: a11 p*1+ a21 p*2= v. Player A receives the same average payoff if the 2nd player uses strategy B2, i.e. a12 p*1+ a22 p*2= v. Taking into account that p * 1 + p * 2 \u003d 1, we obtain a system of equations for determining the optimal strategy S "A and the value of the game v: (3.6) Solving this system, we obtain the optimal strategy (3.7) and the value of the game (3.8) Applying the theorem on active strategies when finding SВ* - the optimal strategy of player B, we obtain that for any pure strategy of player A (A1 or A2) the average loss of player B is equal to the price of the game v, i.e. (3.9) Then the optimal strategy is determined by the formulas: (3.10 )
Pure Strategy- deterministic (excluding randomness) plan of action. In the previous chapter, we considered only pure strategies. Mixed strategies will be discussed in Section 2.2, but for now, unless otherwise stated, by strategy we always mean pure strategy.
Very often in the process of presentation we will illustrate the concepts of solution with examples of bimatrix games, so we will give the corresponding definitions.
Definition 2.1. end game is a game in which the set of players and the sets of strategies of each player contain a finite number of elements. The ultimate game of two persons is called bimatrix game.
The last name comes from a convenient form of recording winnings in such a game - using a double matrix.
For further analysis, it is convenient to divide the strategies in an arbitrary strategy profile s into the strategy of some /-th player s, and the strategies of all other players s_ (. Formally, s = (.y, s,). It is not implied here that we swap the coordinates of the strategy profile , we only introduce another way to denote it.
The first concept of solving the game that we will consider is equilibrium in dominant strategies.
Definition 2.2. The strategy of the /-th player strictly dominated his strategy s" if Uj(s jt s ,) > h,(s", s ,) for any set s , of the strategies of the remaining players. In this case, the strategy s" is called strictly dominated.
Essentially, this means that for any fixed in the set of strategies of the remaining players, the i-th player, choosing a strategy s, obtains strictly bigger win than when choosing a strategy s". It is logical to assume that a rational player should not choose strictly dominated strategies. Such an assumption in the simplest games may be sufficient to find a solution to the game.
Definition 2.3. Strategies Profile s* =(s*, s^,..., s*) is called balance in (strictly) dominant strategies, if for any i-th player the strategy s" strictly dominates any other of his strategies.
It may seem that this concept of solution can only lead to trivial conclusions. Each player has among his strategies one that will give him a payoff more than any other, no matter how his opponents act. Then he will apply exactly this strategy in equilibrium. Everything is pretty obvious. But it is precisely this situation that is typical for, perhaps, the most famous and very important for the analysis of a number of practical situations of the game “prisoners' dilemma”.
Example 2.1 (prisoners' dilemma). The two criminals are in custody in different cells and cannot communicate. The investigation has sufficient evidence to convict each of them for a minor crime for one year. But for a major crime, for which criminals have been facing ten years in prison, the investigation does not have enough evidence. Representatives of the investigation offer each of the criminals a deal: the criminal will receive a term for
one year less if he gives evidence against his partner, which will be enough to charge the latter with a major crime. Assume that criminals are only concerned with the number of years they will spend in prison, each additional year is minus one unit of utility. Then the payoffs of criminals can be represented by the following double matrix:
In the case when the participants in the game are not named, we will assume that the different strategies of the first participant correspond to the rows of the double matrix, and the strategies of the second participant correspond to the columns. If in our example the first prisoner testifies and the second does not testify, then the first will be released, and the second will receive ten years in prison.
It is easy to see that, no matter how the other prisoner acts, the gain is greater (the term of imprisonment is shorter) if you give evidence (for the first player, the first coordinates in the first row of the double matrix are strictly greater than in the second row, for the second player, the second coordinates in the first column double matrix is strictly greater than in the second column). Then the equilibrium in the dominant strategies will be the profile of the strategies (testify, testify).
What is interesting in this example is that the players, choosing behavior that increases their payoff, end up in a situation where their payoffs are low compared to the opposite situation, where both choose to remain silent. The explanation lies in the presence of a strong external effect, i.e. the strong influence of the actions of one player on the payoffs of another player. As a result, the equilibrium profile of strategies turns out to be the only Pareto inefficient one in this game. Note that Pareto efficiency, desirable from the point of view of the participants in the game, may not be desirable from a social point of view, as in this case.
Situations like the Prisoner's Dilemma often occur in the analysis of economic situations. Consider, for example, a competition between two stores selling a similar set of products. For simplicity, let's assume that stores can charge only two price levels - high or low. Consumers naturally prefer to buy from a store with lower prices. Then the payoffs of stores, characterized by their profits, may look, for example, as follows:
From the point of view of equilibrium, the situation here is analogous to the Prisoner's Dilemma - equilibrium in dominant strategies (low prices, low prices) is the only Pareto inefficient profile (and also desirable from a social point of view).
The already mentioned wide popularity of the Prisoner's Dilemma was the reason why, using its example, they tried experimentally to test the correctness of the predictions of game theory. The test was that two strangers it was proposed to play a game for money with prizes (for example, in dollars) close to those indicated for the game of two stores. Each of the participants made a decision separately (often anonymously) and did not know the decisions of the other player before receiving the winnings. It turned out that under such conditions, in many plays of the game, the players did not arrive at an equilibrium result, if we assume that cash prizes correctly estimate their winnings. Of course, it does not follow from the results of these experiments that the predictions of game theory are incorrect, but only that, when evaluating their payoff, the players took into account non-monetary factors - considerations of altruism, fairness, etc. If the payoffs of the players are correctly estimated, then the players should prefer the dominant strategy, and hence choose it (in the spirit of revealed preferences in microeconomics). Therefore, the value of experiments of this kind is not in testing game-theoretic predictions, but in assessing the role of non-material motivation in the actions of individuals.
Significantly less than the concept of strong dominance, game theory uses the concept of weak dominance.
Definition 2.4. The strategy of the /-th player s, weakly dominant his strategy s" if m,(s, s ,) > m ; (sJ, s ,) for any set of other players' strategies s_j, moreover, for at least one set of strategies of other players, the inequality is strictly satisfied. Then the strategy s" is called weakly dominated.
In the case of non-strict inequalities, it is no longer possible to assert that a rational player will not choose a weakly dominated strategy, although such behavior seems quite logical. There is, although rarely used, a definition of equilibrium in weakly dominant strategies analogous to the case of strongly dominance.
Definition 2.5. The strategy profile s* = (s*, Sj,..., s*) is called equilibrium in weakly dominant strategies, if for any i-th player the strategy s" weakly dominates any other of his strategies.
Example 2.2 (closed second price auction). A closed auction of the second price is held among two persons. The auction is arranged as follows. Each of the participants indicates a non-negative rate, not knowing the rates of other participants (in the envelope). Member who made the highest bid, pays the maximum amount among other participants' bids (i.e. the amount of the second but the value of the bid) and receives some item. If, for example, the players' bids were 100 and 90, then the participant who made the bid 100 wins the auction, he acquires the item for 90 - the size of the second bid. Let each participant have an assessment of the subject, expressed in monetary units, v2> 0. These estimates are known to all participants. Let, for simplicity of describing the game, if both participants indicate the same rate, then the object goes to the first participant.
In this game, the strategy of the first player s will be the size of his bet. Since the rate is non-negative, the set of all possible strategies
5, = 0 = u,(o, s 2) > w,(s, s 2) = u, - s 2 v x weakly dominates strategy s,.
We have shown that for the first player, the strategy to name one's score as a bet weakly dominates any other strategy. It is easy to check that a similar statement is true for the second player as well. We note that in our reasoning we never used the fact that the player knows the estimate of the other player, and hence, in the case of a game with incomplete information in closed auction the second price to name your assessment will be no less profitable than to make any other bet.
It may seem that it is unprofitable for the seller to arrange an auction of the second price, when he can arrange an auction of the first price and receive the value of not the second, but the first bid. However, the value of rates in the case of an auction of the first price in equilibrium will be lower. We will talk more about the yield of auctions in Chap. 5. In the meantime, we note that the second price auction is very popular and is widely used, for example, by companies Google and "Yandex" when selling contextual advertising on the Internet.
Equilibrium in dominant strategies exists only in small class games. Typically, players do not have a single strategy that dominates all others. But the concept of dominance allows finding solutions in a wider class of games. To do this, you need to conduct consistent reasoning about the actions of the players. We have already noted that a rational player will not choose a strictly dominated strategy. But this means that the other player can analyze the game, ignoring the possibility of the opponent's choice of such a strategy. Perhaps some analysis will reveal that another player has a dominated strategy that was not dominated in the original game. And so on. Let us give a formal definition.
Process sequential exclusion of strongly dominated strategies is set as follows. Let us exclude all strictly dominated strategies of the players from consideration, i.e. consider a new game in which all dominated strategies are excluded from the set of possible strategies of the players. Then in this new game we eliminate all strictly dominated strategies, and so on.
It is possible that such a process will end when the players have several strategies left, but it is possible that each player will have only one non-excluded strategy, then it is logical to consider a set of these strategies as a solution to the game.
Definition 2.6. If, as a result of sequential elimination of strongly dominated strategies, each player is left with a single strategy, then the profile of these strategies is called dominance equilibrium.
In Example 1.1, we have obtained just such an equilibrium. Let's consider one more example.
The strategy profile (N, P) is the only Nash equilibrium in this game. But note that in order to choose P, the second player must be sure that the first player does not choose B. But the payoff of the first player is the same if the second player chooses II. In addition, by choosing B, the first player may not be afraid that the second player will choose L. Perhaps the rational second player will think about choosing strategy C.
The second question, for which no unambiguous answer has yet been found: how do players come to Nash equilibrium?
The ideal theoretical scenario is as follows. Players independently form expectations about the actions of other players, and then choose the actions that maximize their payoff given the given expectations. If, in this case, the expectations correspond to the actions actually chosen by the players, then we obtain the Nash equilibrium. This line of reasoning allows us to call the Nash equilibrium a situation with self-fulfilling expectations. But where do expectations come from? And which of the Nash equilibria, if there are several, will be chosen as a result of the described process? In the framework of the considered scenario, these questions remain unanswered.
Another approach involves the presence of player training. Players either theoretically learn how to play a given game (imagine students Faculty of Economics), or have experience in similar interactions (for example, an experienced worker comes to new team), which allows them to correctly form expectations and choose the optimal behavior. This scenario helps to explain the formation of expectations, but it, firstly, reduces the scope game models only to standard, studied and frequently occurring situations of interaction, and secondly, it can lead to the fact that situations of single and repeated interaction are not distinguished, and the latter differ significantly in terms of strategies and solution methods within the framework of game theory, which will be discussed in more detail. said in ch. 4.
The third scenario is that there is a prior agreement between the players, or customs, or laws, or instructions from third parties that govern the interaction of the players. In this case, agreements or instructions may not be binding, but if it is recommended to play the Nash equilibrium, then none of the players has the desire (alone) to deviate from the prescribed behavior. It is clear that such a scenario is not possible in every situation. In addition, the very process of forming an agreement or involving third parties can become part of the game.
Finally, the third natural question that arises when studying the concept of Nash equilibrium is the following: is there any empirical evidence that real players usually choose equilibrium strategies? Here again it is extremely difficult to give a short and unambiguous answer. At the same time, the nature of the problems that arise is more consistent with the subject matter of experimental economics. Therefore, we confine ourselves to the recommendation to turn to specialized literature, for example, the book, where the questions of experimental methodology are excellently analyzed and a number of results are presented.
There are games that do not have an equilibrium in pure strategies (see Example 3.1), so the question arises: what conditions are sufficient for the existence of such an equilibrium? Let us formulate and prove the assertion about the existence of a Nash equilibrium in pure strategies in games that are not finite.
Statement 2.3. If the sets of strategies for each of the players S t are non-empty convex compacta in Euclidean space, and the payoff function of each player And- continuous in s and quasi-concave in 5, then the game has a Nash equilibrium in pure strategies.
Proof. Recall the formulation Kakutai's theorems, which we will use in the proof. Let X- non-empty convex compact set in R n , X* is the set of its subsets and/ is such an upper semicontinuous mapping from X V x*, that for each point x e x a bunch of f(x) non-empty, closed and convex. Then the mapping / has a fixed point.
The idea of proving our assertion is to construct a mapping that satisfies the conditions of Kakutani's theorem. To do this, we slightly redefine the display of the best answer. We will, purely technically, assume that the best answer depends not only on the strategies of other players, but also on the player's own strategy s y (s). With a change in the player's own strategy, with the strategies of the other players fixed, the best answer, of course, will not change. Now let's introduce a notation for displaying the best answer for all players as a Cartesian product s(s) = s,(s) x s 2 (s) x... x s n (s). This mapping to each profile assigns a set of profiles in which each player the best way responds to the strategies of other players. The fixed point of the mapping S, i.e. profile s such that s e s(s)> is by definition a Nash equilibrium. Let us show that the mapping 5 satisfies the conditions of Kakutani's theorem. Verification of each condition will constitute a separate point of proof.
- 1. Let us show that the set S all profiles - a convex compact. Since, under the condition of asserting the set of strategies of each of the players S, are non-empty convex compact sets, then the Cartesian product S = S t X S2 X...x S n is a convex compact.
- 2. Display s has non-empty images. By the Weierstrass theorem, the continuous function And- reaches on a closed bounded set 5, its own maximum value. Hence, s has non-empty images.
- 3. Display images s closed and convex. Since the payoff function of each player u t quasi-concave in s if then, by the property of a quasi-concave function, the set $. = (s. | u t (s i9 s .) > k) for fixed s .and k is closed when the domain of definition is closed and is convex if it is not empty. Since this is true for any k, then it is also true that the set 5. = (5/1 u t(s", 5 ,) > maxw.(s., s .)}
convex. But then the Cartesian product 5(5) = s x (s) X s2(S) x... x s n CS) is closed and convex.
4. Let us show that the mapping § semi-continuous from above. We use the continuity condition for the function And, by s. We will prove by contradiction. Let's assume that the display § ns is upper semicontinuous. Then there are sequences of strategy profiles s m And s m , Where T - sequence element number, such that for any T s""e S, s m e s(s""), lim s"" = s° e S, but lim s"" = s° g lim s(s""). This means that there is a
t~* oo t->/And -? oo
rock for which strategy s f ° is not the best response to s 0 , i.e. there is a strategy s" such that and,(s", s 0 ,) > u,(s] s°;). Then one can find e > 0 such that m,(s/, s 0 ,) > m,(s ; °, s 0 ,) + Ze, whence
Since, by assumption, the function m is continuous, lim s m = s°, lim s"” = s°,
m*oo m-*oo
with a large enough m right
Combining inequalities (2.8)-(2.10) into one chain, we obtain
It follows from relations (2.11) that u,(s", s"") > m,(s/", s"") + s, but this contradicts the condition s"" e s(s""), since s" gives a strictly greater payoff than s/", in response to s"". They came to a contradiction. Therefore, our original assumption that s is not upper semicontinuous was wrong.
We have shown that the mapping S satisfies all the conditions of Kakutani's theorem, and hence has a fixed point. This fixed point is the Nash equilibrium. Assertion 2.3 is proved. ?
Statement 2.3, in particular, guarantees the existence of a Nash equilibrium in Example 2.7, but not in Example 2.8, where the payoff functions of the players are discontinuous.
"Example from work.
The player's choice of an action is called move. There are moves personal(the player consciously makes a decision) and random(the outcome of the game does not depend on the will of the player). The set of rules that determine which move a player must make is called strategy. There are strategies clean(nonrandom player decisions) and mixed(the strategy can be considered as a random variable).
saddle point
IN game theory S. t. ( saddle element) - This largest element column game matrices, which is also the smallest element of the corresponding row (in two-person zero-sum game). At this point, therefore, the maximin of one player is equal to the minimax of the other; S. t. there is a point equilibrium.
Minimax theorem
The minimax strategy is called minimax strategy.
The principle that dictates to players the choice of the most "cautious" maximin and minimax strategies is called minimax principle. This principle follows from the reasonable assumption that each player seeks to achieve the opposite goal of the opponent.
The player chooses his actions, assuming that the opponent will act in an unfavorable way, i.e. will try to harm.
Loss function
Loss function is a function that, in the theory of statistical decisions, characterizes the losses due to incorrect decision making based on the observed data. If the problem of estimating the signal parameter against the background of interference is being solved, then the loss function is a measure of the discrepancy between true value estimated parameter and parameter estimate
Player's Optimal Mixed Strategy is a complete set of applications of his pure strategies in multiple repetitions of the game under the same conditions with given probabilities.
A player's mixed strategy is a complete set of application of his pure strategies in the case of multiple repetitions of the game under the same conditions with given probabilities.
1. If all elements of a row are not greater than the corresponding elements of another row, then the original row can be deleted from the payoff matrix. Likewise for columns.
2. The price of the game is unique.
Doc-in: let's say there are 2 game prices v and , which are achieved on the pair and respectively, then
3. If the same number is added to all elements of the payoff matrix, then the optimal mixed strategies will not change, and the price of the game will increase by this number.
Doc-in:
, Where 
4. If all elements of the payoff matrix are multiplied by the same number that is not equal to zero, the price of the game will be multiplied by this number, and the optimal strategies will not change.
