Symmetric Game Consider the following -person game. Each player has a strategy which is a number x (0 x 1), thought of as the player s contribution to the common good. The net payoff to a player playing x whose partner contributes y has the form: P(x, y) = B(x+y) C(x) where B is the benefit to the player and depends on the sum of the two contributions, and C is the cost to the player and depends on his own contribution. For example, if the strategies of the two players are 0.3 and 0.5, then the first player gains and the second player gains P(0.3, 0.5) = B(0.8) C(0.3) P(0.5, 0.3) = B(0.8) C(0.5). The first player will gain more as he contributed less whereas both players have the same benefit. This type of cooperative game is much studied in both economics and biology because it serves to give us a fundamental understanding of interactions between individuals and organisms in the struggle to do well within a social community. In animal behaviour a typical realization involves two parents balancing their individual investment in the common brood. Now the objective of the game is to get as large a payoff as possible. So the question for me as a player is, what s the best value of x to play? Of course, that will depend on my partner s y because my payoff depends on both our strategies. So do I know what my partner s y is going to be? Well maybe I will and maybe I won t. Maybe I ll have some information about it, but not enough to know exactly what it will be. There are different situations here that are worth exploring. Let s begin with the assumption that I know my partner s y. Example 1. Take the benefit and cost functions: Thus: B(z) = z(4 z) C(x) = x. P(x, y) = (x+y)(4 x y) x. Suppose I know the contribution y of my partner. What is my optimal contribution x*? Find a general expression for x* in terms of y. I come to class with a computer with MAPLE running and I tell the students that I will plot for them any graphs that they like, or for that matter do anything else that lies within the power of MAPLE (e.g. solve hard equations). All they have to do is ask. I also warn them about the dangers of just blindly taking derivatives and setting them to zero. Work from a picture! I exhort. Always insist on seeing what s happening. Array the possibilities. Ask me to plot some graphs. symmetric game 1
They ask me to plot the payoff function. What do you mean? The payoff P is a function of both x and y. [Perhaps they want the contour lines of P(x, y), but no, they re not quite ready for that.] I give them a number of payoff curves for various partners. I give them the curves for 11 different equally spaced y-values. Each curve is the graph of P(x, y) for a particular fixed y. The lowest curve (passing through the origin) is the graph of P(x, 0), the next is the graph of P(x, 0.1), etc. Every second one of these is labeled with its y-value. For example, P(x, 0.8) is my payoff, in terms of my strategy x, if my partner plays y=0.8. This is the graph p = P(x, 0.8) = (x+y)(4 x y) x = (x+0.8)(3. x) x. = x +.4x+.56. This is a parabola and the strategy x giving me the highest payoff is at the vertex of the parabola, where the derivative is zero: dp d = ( x +.4x+.56) dx dx = 4x +.4 = 0 x = 0.6 My resulting payoff is P(0.6, 0.8) = 3.8. This is illustrated in the graph at the right. The graph displays a couple of significant patterns. First of all, the higher is y, the higher is my payoff. That makes sense higher y means higher benefit. The second pattern is that the higher is y, the lower is my optimal contribution x*. That s because as we go up, the vertex of the parabola shifts to the left. That pattern is not at first so obvious. symmetric game
Working with the equations. We want a general expression for x* in terms of y. Now x* = x*(y) is the x-value of the vertex of the y-curve. In terms of the derivative, it is the x-value at which the slope of the y-curve is zero: Solve for x and we will get x*: d d P(x, y) = [(x+y)(4 x y) x ] dx dx = (4 x y) (x+y) x = 4 4x y = 0 4 y = 4x x* = y This is the general expression we are after. As a check, for the y=0.8 curve, it gives us the vertex at: as we found above. x*(0.8) = 0.8 1. = = 0.6 Note that we are taking the derivative here with respect to x with y held constant (giving us a fixed payoff curve). In this case, we typically use the partial derivative notation to signal that one of the two variables is being held fixed with the other treated as the independent variable. The equation would then be written: (x+y)(4 x y) x = 0. x Example. The evolution of the game. Suppose we have a large population of individuals who mix with one another and play this game repeatedly, generation after generation, each time with randomly chosen partners. Different individuals of course might have different strategies, so payoffs will vary. Suppose that I never know the strategy of my current partner (until after I ve committed myself to an x-value) but I do know (from experience) the configuration of different strategies played by the population as a whole. Suppose that players who have higher payoffs on average do better in life and live longer and have more offspring, so individuals are often examining their past performance and possibly improving their strategy. And of course offspring tend to imitate the strategies of their parents especially when these strategies are working well. All this means that over time strategies with an above average payoff will increase in frequency and strategies with a below average payoff will decrease in frequency, and the population strategy mix will change or evolve. Here s the question. Suppose we start with a population with a fairly even mix of all possible strategies. How will the population evolve? What will it look like after a long time? symmetric game 3
As usual it s important to start with a picture and the graph from Example 1 is drawn at the right. It generates a good discussion. It is observed that in a population where most players use low values of y, say 0.1, 0., the players who do best are those few who use high values, say x = 0.8 or 0.9. Conversely in a population where most players use higher values of y, say 0.8, 0.9, the players who do best are those few who use lower values, say x = 0.5 or 0.6. How will such a population evolve over time? In a low population, higher strategies do better and in a high population lower strategies do better. Perhaps the strategy mix will converge to some intermediate value. Perhaps at such a point, those who use an x which is too high or too low will do less well than the population average. Such a payoff pattern should stabilize the population at the intermediate value. What should that intermediate value be? After some thought, the students suggest that it should be the y-curve whose vertex occurs where x=y. That will guarantee that every other value of x does less well than does y itself. How do we find this? Finding the equilibrium y. The equilibrium y is the value of y for which the vertex of the y-curve is at x=y. We can get a geometric estimate of this from the picture. The vertex of the y = 0.6 curve has x-value which is definitely greater than 0.6 The vertex of the y = 0.7 curve has x-value which is definitely less than 0.7 So the equilibrium y would seem to lie between 0.6 and 0.7. To pin it down precisely, we ll use the calculus. The equilibrium value of y will be that value for which the vertex of the y- curve occurs at x=y. That is: x*(y) = y. y = y y = y = 3y y = /3. Need picture here of y-curve for y = /3 with x* = /3 marked. This is the equilibrium value of y. It lies between 0.6 and 0.7 as our geometric investigation predicted. This is the y-value we might expect the population to settle down to. To be more explicit, in a population in which everyone plays y=/3, individuals who decide to deviate from the norm do less well than normal (because the vertex of the curve is at x=/3) and they and their offspring will have a greater chance of being eliminated from the population. Because of this property (that when /3 is the norm, then deviants do less well than normals) y = /3 is called a Nash equilibrium named after John Nash, Nobel prize winner and subject of the recent movie Beautiful Mind. symmetric game 4
Asymmetric game We re going to play the same game as before, but this time there are two types of players, type A and type B and each play of the game involves one A and one B. We will think of A as the female and B as the male. Now here s the point the rule is that A has to go first, that is A chooses her strategy before B chooses his, so that when B makes his choice he knows what A has chosen. Because we have two types of players, I ll use new notation for the two strategies used. Let a be the strategy of A and b be the strategy of B. We ll use the same payoff as the previous section. A s payoff is: P(a, b) = B(a+b) C(a) = (a+b)(4 a b) a and B s payoff is P(b, a) = B(b+a) C(b) = (b+a)(4 b a) b. Now the question is, will this change how either A or B plays the game? In particular, does this asymmetry give one or other of the players an advantage? If so, which is it, A who goes first, or B who goes second? Well we have two questions to answer what will A do and what will B do? Of the two, it seems that the second is more straightforward. A plays a cost Ca ( ) and then B plays b cost Cb ( ) common benefit Ba+ ( b) Who has the advantage the female who goes first or the male who goes second? It might seem at first that the male who goes second should be better off as he knows what the female has played, whereas she must act before he has moved. But the female does have some information. She knows that the male will act to maximize his payoff. Does that help to make up for her disadvantage? B s problem. B makes his decision knowing a, so he will clearly choose the value of b which gives him the highest payoff given a. That is, for a given a, B will choose b to: maximize P(b, a) = B(b+a) C(b). This is exactly the problem we had in Example 1 of the symmetric game, with x instead of b and y instead of a. The value of b that maximizes this is: b = x*(a) = a The graphs at the right belong to Example 1 of the previous section with B(z) = z(4 z) C(x) = x Each curve belongs to a different value of y, starting at the bottom for y = 0, through y=0.1, y=0., etc. all the way to y = 1 at the top. After A has played a, B will fasten attention on the correct y=a curve and then choose b to give himself the vertex of the curve.. asymmetric game 1
In the diagram at the right, we suppose that A chooses a=0.6. Then B would choose to maximize his payoff. b = x*(0.6) = 0.7 A s problem. What about A? What does she know that might help her get a good payoff? Well, actually she knows quite a bit. She knows that B wants to maximize his payoff, so that given her a, he will choose b = x*(a) So for any potential a she might play, she knows B s response and she can therefore work out her own payoff! Now her own payoff is P(a, b), which, given that b will be given by x*(a), will be P(a, x*(a)). To maximize this, A wants to choose a to maximize P(a, b) = P(a, x*(a)) = B(a+x*(a)) C(a). That looks a bit complicated, but it s just a function of a. If we plug in the formulas for B and C and x*, we get P(a, x*(a)) = B(a+x*(a)) C(a) = (a+ x*(a))(4 a x*(a)) a a a = (a+ )(4 a ) a = ( a + a)(8 a + a 4 a = 1 [(a+)(6 a) 4a ] 4 = 1 [ 5a + 4a + 1] 4 Now A wants to maximize this, so she sets the derivative to zero: d [ 5a + 4a + 1] dx = 10a + 4 = 0 a = 0.4. A should play a = 0.4. We can then work out B s strategy. He should play b = x*(a) = x*(0.4) = 0.4 = 0.8 The payoffs are: A: P(a, b) = P(0.4, 0.8) = (1.)(4 1.) (0.4) = 3. B: P(b, a) = P(0.8, 0.4) = (1.)(4 1.) (0.8) =.7. That s interesting: look who has the higher payoff A. Evidently it s an advantage to go first. These results should be compared with the results of the symmetric game in Example of the previous section. There, the equilibrium contribution was x = /3 giving a payoff (for both players) of 3.11. When we introduce the asymmetry look what happens! A s contribution gets smaller (0.4) and B s contribution gets greater (0.8). As a result, A has a greater payoff (3.) and B has a smaller payoff (.7). Would you have predicted that? asymmetric game
A contour line game analysis Here we study the game of the last two sections except I m going to give you the payoff information in a new way. Recall that the payoff to a player with strategy x whose partner had strategy y is the benefit of the total contribution minus the individual cost: P(x, y) = B(x+y) C(x). However, instead of giving you a formula for P(x, y) or the family of graphs of P against x (with y-fixed for each graph), as I did in the previous sections, I m going to give you the contour lines in the x-y plane of the function P(x, y), spaced at intervals of P = 0.1. Your job is to analyze the game using this graph first to investigate the long-term behaviour of the symmetric game, and secondly to solve the asymmetric game. That is, you are to do the same analysis as the previous sections, except with the information presented in this new way. P = 1 P = P = 3 The contour lines of P(x, y) This graph displays the payoff function for us in a very different way from the graphs we ve been working with. The heavy contours are the curves P = 1, and 3; the light contours are the intermediate curves spaced at intervals of 0.1. For example, the curve marked P=3 passes through all points (x, y) for which P(x,y) = 3. The curve next curve down from that is the set of points for which P(x,y) =.9. Etc. One way to think about this graph is to interpret P(x,y) as the height of land at latitude x and longitude y, and then this is a topographical contour map of the region. If I walk along one of the curves, I stay level. At any point, the land rises most steeply in a direction perpendicular to the curves. a contour line analysis 1
Example 1. The symmetric game. How might we use this graph? Let s take a particular case. Suppose I am playing the game and my partner happens to play y = 0.5. What are the possible payoffs I might obtain? How are they determined by my x-value? What value of x will maximize my payoff? How does the graph answer these questions for me? Well, the relevant points on the graph are those for which y = 0.5 and this is a horizontal line at height 0.5. Different x s will put me at different points along that line, and we can use the level curves to read off the P-value at each of these points. Which x will give the highest payoff? That will be the point on the line that meets the curve with the highest possible P- value. Since the curves increase in P-value as they get higher, we want the curve that s as high as possible in the sense of being high on the page. That particular curve does not appear on the diagram, but by interpolation we can get a pretty good idea how it will run. Its P-value will be just below 3.9, say 3.88, and it will be tangent to the line at an x close to 0.75. Now how might we use the graph to solve the evolutionary problem? What ought to happen to the population over time? Well the graph above shows that in a population where most everyone is playing 0.5, individuals who play close to 0.75 will have an advantage. So we expect individuals to increasingly make higher contributions. But suppose we have a population where the typical contribution is high, say 0.9. The graph at the right shows that the highest fitness will be obtained by individuals who use intermediate strategies around 0.5 or 0.6. Over time the population strategy should evolve to a point y at which the individuals playing y are doing better than individuals who might come along playing any other x. What does that mean? That the line at height y should be tangent to a level curve at the point x=y, that is the x-value should be the same as the y. This can be represented geometrically in a very fine way. The above condition tells us that the point of tangency should lie on the diagonal y=x. This diagonal is drawn at the right. We want the level curve that has its horizontal tangent at a point on this diagonal. The precise level curve with this property is not in the picture, but by interpolation, we can see that that happens somewhere between x=0.6 and x=0.7, in fact closer to 0.7. By using additional level curves, we would find the point to be close to x=y=/3, the answer we obtained from the analysis of the previous section. a contour line analysis
Example. The asymmetric game. Find a geometric solution to the asymmetric game using the contour line diagrams. Start with the contour line graph regarded as the level curves of B s payoff. So we locate b on the x-axis and a on the yaxis. Now B s behaviour is simple enough and has just been analyzed in Example 1. Given A s choice of a, B will choose the level curve that is tangent to the line y = a and the point of tangency will determine b. This is illustrated in the diagram at the right for a = 0.5. B will play b = 0.75. Given this, what should A play? For example, is the play illustrated at the right optimal for A? Should A play higher than a = 0.5? Or lower? What A has to do is to get hold of her payoff in terms of her choice of a. Now whatever value of a she plays, the diagram will tell her how B will respond, and that will give her a (b, a) point. The set of all those (b, a) points is easy to describe it s the set of all vertices of the family of level curves. Now if we connect up all those vertices, the ones that are shown on the diagram as well as those for all the missing level curves, we ll get a locus which will probably itself be a smooth curve. Using the vertex points that are actually on the diagram, I have drawn this curve at the right. Interestingly enough, it seems to actually be a straight line. How did I draw it that curve? What expression did I plug into MAPLE? In fact for every a, the corresponding optimal b value is what we have called x*(a), and we even have a formula for it: a b = x*(a) =. That s a linear expression so that the curve passing through the vertices of the level curves is indeed a straight line. Okay. That straight line represents the set of options that are available to A. Her job is to decide which of these will give her the largest payoff. How does she use the diagram to get hold of her payoff? How she compare her payoff at different points on that line? Well, she d need the level curves of her payoff function. Isn t that what the diagram shows? No! The level curves in the diagram belong not to her payoff function but to B s. How can she find her level curves? How do we compare B s payoff at different points on that line? That s actually a nice problem and it can be solved with an elegant construction. Don t turn the page until you ve treated yourself to a bit of exploration. a contour line analysis 3
How A compares payoffs at different points. There are a couple of ways she might do this. One is to use the axes as we already have them (with a as the vertical) and plot A s level curves sideways using the vertical axis as the independent variable. But that involves replotting the level curves. Is there a way of using the level curves that are already there? Well we could also simply switch the axes, letting A have the horizontal axis and B the vertical. Then the plotted level curves certainly belong to A. But we still need to see the line b = x*(a), because that tells A how B will respond to her and so it displays her options. Can you see how to replot that? Okay. Let s switch the axes. Now the level curves belong to A s payoff. But that s still the original line which now (because of the switched coordinates!) has equation a = x*(b). To recover the line we want, which is the line b = x*(a) we have to reflect the line in the diagonal b = a. The line and the reflected line are displayed at the right. Finally we do the analysis. First we remove the irrelevant line (the original b = x*(a)) and then we study the resulting diagram. The curves are the level curves of A s payoff, rising in value as we go up. The line on the diagram is the new line b = x*(a), and it is the line of A s feasible points. For different values of a the height of that line gives B s response, so these are the points A must pay attention to. Which point on this line will give her the highest payoff? Well what is the highest level curve that intersects the line? It s the one that is in fact tangent to the line. That s the point marked with a = 0.4 and b = 0.8 as we calculated before. A will conclude that she should play a = 0.4. B will then respond with b=0.8 (to maximize his payoff) giving A the payoff P(a, b) = 3. as seen from the diagram. a contour line analysis 4