Dialectical Equilibrium

Yuke Li ∗ Weiyi Wu †

April 15, 2015

1 Introduction

Many applied research in social sciences has benefited from the advancement of game theory tools. Among

the tools, those that are prevalently used include but are not restricted to: 1. the class of games with finite

strategy space; 2. the existence theorem of mixed strategy Nash Equilibrium; 3. the assumption of complete

rationality; 4. the class of infinite-horizon games in discrete time.

However, these commonly used tools fall short of providing a basic framework to describe a class of

human behavior prevalent in many spheres. The behavior could be noncooperative, cooperative or both.

First, people may have infinite choices of possible actions but only have limited time frame to react. People

are only able to continuously “move” to another strategy, as required by the continuity of the time and

space we live in. Second, by the same reason, people do not usually have full-vision of their infinite-strategy

space. The kind of strategies chosen by players depend on the point of the time players arrive at. Thus, the

whole strategy space gradually “unfolds” itself to players over time. An issue arises when people’s visions

about the strategy space differ. This might result in those with farsightedness dominating those without

it. Third, in many real life situations, people decide and execute the actions almost concurrently. They are

able to learn about the other people’s pure-strategy actions in the previous point of time, and then adapt

accordingly for the rest of the time.

These features have important implications for the modeling of the strategic situation. Our research

examines these behavior less explored in the current game-theoretic literature but prevalent in real life. To

model this class of behavior, we apply the theory of dynamic system in continuous time and on infinite

strategy space. The other features of the modeling are: 1. the number of players is not limited; 3. the game

could be noncooperative, cooperative or both; 4. the equilibrium concept applies to pure strategy; 5. players

may have non-full vision and be shortsighted about the strategy space.

Without farsightedness about the strategy space, the equilibrium concept should be at least “local” in

feature. We therefore claim that the analysis of this class of behavior require a new equilibrium termed

as dialectical equilibrium. The term is from (Hegel, 1977; Horkheimer and Adorno, 2002), because certain

competitive behavior from the perspective of dynamic system could approximate “thesis and antithesis”. This

term is borrowed only for convenience and implies no more philosophical meanings. The basic framework is

called Dialectical Game.

∗Department of Political Science, Yale University, New Haven, CT, 06511†Department of Computer Science, Yale University, New Haven, CT, 06511

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The game can predict some interesting phenomenon the current literature has not spoken to. For instance,

the assumption on visions shares some conceptual resemblance to the “level-k reasoning” (Crawford and

Iriberri, 2007) in the bounded-rationality literature. But it can predict inter-temporal manipulation of the

others, which is what the “level-k reasoning” has not predicted.

2 Preliminary

2.1 Glossary of Dynamical System

I give a brief glossary of key concepts to be used here. (Coxeter, 1998)

Attractor An attractor is a set of numerical values a system evolves towards with different initial

conditions. System values close enough to the attractor values are rubout to slight perturbations.

Attractive Fixed Point An attractive fixed point of a function f is a fixed point x0 of f such that for

any value of x in the domain that is close enough to x0, the iterated function sequence, x, f(x), f(f(x)),

f(f(f(x))), . . . converges to x0.

Limit Cycle A limit cycle is an isolated closed trajectory, which means the neighboring trajectories

either spiral rowards or away from the limit cycle.

2.2 Set Up

The elements of the game are: the game has N players, with N being any positive integer. The game takes

place in continuous time t ∈ [0,+∞).

At each t, each of the players is in a certain state si. They each choose an action concurrently. The state

is defined on compact spaces. The states are determined by players’ actions and payoff relevant.

The actions each of the players choose ai is a derivative of the state they are in. ai = ∂tsi. This is

different from a classic setting of stochastic games, where the actions are defined separately from states. In

our game, what the players choose are a velocity. And velocity is the derivative of the states.

Denote the utility of each player i as ui(s⃗). For each player, his goal is to maximize the average utility

over time. The maximization problem is:

limT→+∞

1

T

∫ T

0

u(s⃗(t))dt

With full knowledge of his own strategy space, the others’ strategy space, utility functions, current and

future actions, a rational player can be able to determine the optimal action at each t to maximize the

average utility over time.

However, in reality, first, the players generally have no such full knowledge. Second, they are not able to

precisely calculate the optimal action in real time. Thus, a reasonable assumption about human behavior is

that : without imposing the “global” vision and rationality, they would behave as in a “greedy-algorithm”

manner — select a action for them within a small time interval, rather than over the whole time span. More

precisely, they will only select the action based on the gradient of the utility functions. Then we can assume

their strategies are bounded by this gradient. ai(t) = λ∂siui(s⃗(t)), where 0 ≤ λ ≤ 1.

As will be defined, a dialectical equilibrium is an attractor.

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2.3 A two-player example

Although the game is applied to an unlimited number of players, we illustrate a two-player version to convey

the basic ideas.

Player 1 and Player 2 play a game. Player 1’s action is to control the x coordinate and his direction of

action is either left or right; player 2’s action is to control the y coordinate and his direction of action is

either forward or backward. They have independent utility based on the coordinates (x, y).

(a) Utility for i (b) Utility for j

Figure 1: Both Utilities

(a) Local maxima (green solid line) and local minima(green dashed line) for i

(b) Local maxima (green solid line) and local minima(green dashed line) for j

Figure 2: Local maxima and local minima for both

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The local maxima for both are shown in Figure 2. The intersection of the local maxima for both are the

local Nash Equilibria. When the coordinates reach the local maximum for either, that player does not have

a direction to increase utility immediately. If the coordinates reach the local maximum for both, this is a

steady state where both are satisfied.

3 The Game as a Dynamical System

Suppose the utility function for player 1 is u1(x, y)1, and the utility function for player 2 is u2(x, y). The

system is time invariant. Now assume that the players here has only minimal rationality — they do not

pursue a maximum because they cannot think that far. But instead change their strategy based the gradient

of the utility function shown below {∂tx = ∂xu1

∂ty = ∂yu2

The partial differential equation describes how their actions change over time. The possible trajectory of

coordinates is shown in the phase portrait in Figure 3.

Figure 3: Phase Portrait

The intersections of the local maxima are local Nash equilibria of the game. Thus, by the definition of

Nash equilibria, they are one type of fixed points, attractive fixed points, for the dynamic system.

For the game, there are the other types of attractors, which all fall into dialectical equilibrium. For

a two-dimensional non-ergodic time-invariant system such as this one, there are two types of attractors.

The first is attractive fixed point. The second is limit cycle. For ergodic systems, there are no attractors

1Suppose u1 is infinitely differentiable for now.

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obviously. 2

3.1 Types of Attractive Fixed Points

The conditions for fixed points are ∂tx = ∂ty = 0, which is equivalent to ∂xu1(x, y) = ∂yu2(x, y) = 0.

Using phase plane methods, we can check whether these fixed points are attractors. That is by linearized

approximations near the extrema that satisfy the conditions for being fixed points.

Let the coordinates of a point satisfying the fixed point conditions be (x0, y0). The linearized approxi-

mation is in the form {∂tx = A(x− x0) +B(y − y0)

∂ty = C(x− x0) +D(y − y0)

The points that satisfy the conditions can be classified based on the values of p, q and ∆ defined as below,

for which the points would have different properties and denote different things. As will be discussed, the

commonly known Nash equilibria (global or local) correspond to the case where p < 0, q > 0 and ∆ ≥ 0.

The below graph is from (Jordan and Smith, 2007).

2p − 4q=

= 0q

> 0

< 0

> 0

< 0

p

= Ax + Bydxdt

= Cx + Dydydt

q = AD − BCp = A + D

Figure 4: Different Extrema

Some examples of different fixed points are given below.

3.1.1 p < 0, q > 0 and ∆ ≥ 0 (A Global NE and An Attractive Fixed Point)

The utility function for them are

2For non-ergodic systems, the attractors not only consist of local Nash equilibria. In particular, attractive fixed points arenot necessarily Nash equilibria.

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{u1(x, y) = −x2

u2(x, y) = −y2

Near (0, 0), the linearized approximation is {∂tx = −2x

∂ty = −2y

And the values of p, q and ∆ are p = −4

q = 4

∆ = 0

Figure 5: A Global NE and An Attractive Fixed Point

3.1.2 p < 0, q > 0 and ∆ < 0 (An Attractive Fixed Point)

The utility function for them are{u1(x, y) = −(x+ y)2 − (x− 2y − 2)(x− 2y − 1)(x− 2y + 1)(x− 2y + 2)

u2(x, y) = −8(2x− y)2

Near (0, 0), the linearized approximation is

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{∂tx = 8x− 22y

∂ty = 32x− 16y

And the values of p, q and ∆ are p = −8

q = 576

∆ = −2240

Figure 6: An Attractive Fixed Point

Note that this fixed point is not a Nash equilibrium as the intersection of local maxima. It is a dialectical

equilibrium because it is the intersection of local minima and local maxima.

3.1.3 p > 0, q > 0 and ∆ < 0 (Not an Attractive Fixed Point)

The utility function for them are{u1(x, y) = −(x+ y)2 − (x− 2y − 2)(x− 2y − 1)(x− 2y + 1)(x− 2y + 2)

u2(x, y) = −(2x− y)2

Near (0, 0), the linearized approximation is{∂tx = 8x− 22y

∂ty = 4x− 2y

And the values of p, q and ∆ are

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p = 6

q = 72

∆ = −252

Figure 7: Not an Attractive Fixed Point

Note that this intersection is not an attractive fixed point and therefore not an attractor. It is also not a

dialectical equilibrium. Though all the extrema happen at the same coordinates as in the previous example,

the ratios of the two players’ utility gradients in the previous example cause that fixed point to be attractive.

And the ratios of the two players’ utility gradients here cause this fixed point to be repelling.

3.1.4 p < 0, q < 0 and ∆ > 0 (Not an Attractive Fixed Point)

The utility function for them are{u1(x, y) = −(x+ y)4 − (x− y − 2)(x− y − 1)(x− y + 1)(x− y + 2)

u2(x, y) = −8(4x+ y)2

Near (0, 0), the linearized approximation is{∂tx = 10x− 10y

∂ty = −64x− 16y

And the values of p, q and ∆ are

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p = −6

q = −800

∆ = 3236

Figure 8: Not an Attractive Fixed Point

This fixed point is not an attractive fixed point. It is a saddle point.

3.2 Limit Cycles

The limit cycle is another type of attractors and dialectical equilibrium. Using the limit set methods, some

examples are given below.

3.2.1 p > 0, q > 0 and ∆ < 0 (Van der Pol equation)

The utility function for them are u1(x, y) = xy

u2(x, y) =1

2

(1− x2

)y2 − xy

Near (0, 0), the linearized approximation is{∂tx = y

∂ty = −x+ y

And the values of p, q and ∆ are

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p = 1

q = 1

∆ = −3

Figure 9: Different Steady States

3.2.2 p = 0, q > 0 and ∆ < 0 (Duffing equation)

The utility function for them are {u1(x, y) = xy

u2(x, y) = −xy − x3y

Near (0, 0), the linearized approximation is {∂tx = y

∂ty = −x

And the values of p, q and ∆ are p = 0

q = 1

∆ = −4

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Figure 10: Different Steady States

3.3 Why Nash Equilibrium Fails in Certain Situations?

If an attractor is beneficial for both, both will converge onto the attractor. An obvious case is the first

example of attractive fixed points, where p < 0, q > 0 and ∆ ≥ 0.

However, if the attractor is beneficial for only one player, such as the second example of attractive fixed

points, the benefited player 2’s action could match that of player 1. By the property of their strategies being

bounded by the utility gradient, (∂tx, ∂ty) will be always kept as the best possible direction for player 2.

Then player 2 might be able to engage in intertemporal manipulation of player 1.

References

Coxeter, Harold Scott Macdonald. 1998. Non-euclidean geometry. Cambridge University Press.

Crawford, Vincent P and Nagore Iriberri. 2007. “Level-k Auctions: Can a Nonequilibrium Model of Strate-

gic Thinking Explain the Winner’s Curse and Overbidding in Private-Value Auctions?” Econometrica

75(6):1721–1770.

Hegel, Georg Wilhelm Friedrich. 1977. “Phenomenology of Spirit. 1807.” Trans. AV Miller. Oxford: Oxford

UP .

Horkheimer, Max and Theodor W Adorno. 2002. Dialectic of enlightenment: Philosophical fragments. Stan-

ford University Press.

Jordan, Dominic William and Peter Smith. 2007. Nonlinear ordinary differential equations: an introduction

for scientists and engineers. New York.

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