# Strategic Islands in Economic Games: Isolating Economies From Better Outcomes

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## Abstract

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## 1. Introduction

## 2. Game Theory

_{c}(left) and p

_{c}(right) are probabilities over col’s two choices and p

_{r}(top) and p

_{r}(bottom) are probabilities over row’s choices. Each player then seeks to independently choose the probabilities that maximise their expected payoffs given by:

_{c}(left) = 1. A mixed strategy is one in which players choose using a distribution different from a pure strategy. While the Nash equilibrium concept is not the only form of rationality that players may follow it has been a cornerstone of modern micro-economic game theory since it was first proposed. An alternative called the Quantal Response Equilibrium will be introduced later.

_{r},Q

_{c}∈ [−1, 1], given by the mappings: ${p}_{c}(\text{left})\to {\scriptstyle \frac{1-{Q}_{c}}{2}},{p}_{c}(\text{right})\to {\scriptstyle \frac{1+{Q}_{c}}{2}},{p}_{r}(\text{top})\to {\scriptstyle \frac{1-{Q}_{r}}{2}}$ and ${p}_{r}(\text{bottom})\to {\scriptstyle \frac{1+{Q}_{r}}{2}}$. This allows us to express the expected utilities in the following slightly more compact fashion. First construct four functions (explained next) for each player based upon their respective utility matrices in matrices 1 where x ∈{r, c}:

_{r}and Q

_{c}composed of an intrinsic term, a self-interaction term, an influence term from the other player and a joint interaction term:

_{r}and s

_{c}:

## 3. The Quantal Response Equilibrium for Game Theory

_{i}p(i) ln(p(i)) which has often been interpreted as a measure of the spread or diversity of p. This interpretation was discussed at least as early as 1975 in economics [18] and it is a well established measure in ecology [19] in which understanding the biodiversity of an ecological system is important. In combination with other desiderata [20] the entropy is the unique function that has the property that when it is applied to independent subsystems the sum of the parts is equal to the whole. For example if a system can be partitioned into two independent subsystems A and B (as we do below) with probability distributions p and q with joint probabilities pq, then S(A+B)= S(A)+S(B) and S(pq)= S(p)+S(q), we expand on this next.

_{x})= −∑

_{i}p

_{x}(i) ln(p

_{x}(i)) of each player’s distribution. For a 2 player game in which each player has k discrete choices from which to select, each player individually maximises the entropy associated with the distribution over their choices:

_{x})and then solving the stationary solutions ∂

_{{}

_{px}} (p

_{x})=0 for p

_{x}in the following way:

_{r,c}(i, j)= p

_{r}(i)p

_{c}(j), which is the assumption of independence of non-cooperative game theory. This should be compared with the Nash equilibrium in which individuals independently maximise their utilities, not their entropies, via the expected utility functions. The difference can be interpreted in terms of individuals vs. large populations, discussed below in terms of the Fokker–Planck equation. Equation (21) can be rewritten in the form:

_{r},β

_{c}) pair can each separately and independently vary from ∞ (perfect rationality: when (β

_{r},β

_{c}) = (∞, ∞) the Nash equilibria are recovered), through 0 (strategic indifference: when (β

_{r},β

_{c}) =(0, 0) the players play uniformly across strategies) to −∞ in which case players choose strategies that harm their expected utilities. With this range of parameters, the β term describes the rationality [23] or even the persona [24] of each player. In this sense it can be thought of as a psychological parameter that mediates the decision-making process between the accumulation of information regarding the game and the behavioural outcome Q

_{x}. It should be emphasised that every (β

_{r},β

_{c}) pair of Equations (22) and (23) results in an equilibrium and variation of these parameters results in another equilibrium point. Even in the simplest case of two players and two choices, there is the possibility of non-trivial results; the following sections discuss some interesting examples. Empirical evidence for the QRE as a model of individual choice can be found in the original McKelvey and Palfrey paper [17] as well as in 2×2 games such as the matching pennies game and its generalisations [25].

#### 3.1. Quantal Equilibrium Paths: Prisoners’ Dilemma

_{x}= −1), right or bottom = accusing the other person (Q

_{x}= +1):

_{r,c}(U

_{x})=0 and the intrinsic utility is that of a hostile environment: f

_{i}(U

_{x})= −1.5 so that if both players played uniform strategies: Q

_{x}=0, the expected utility would be negative. Both players have an incentive to change their behaviour Q

_{x}to accuse the other person: ${\scriptstyle \frac{\partial E({U}_{x})}{\partial {Q}_{x}}}>0$, no matter what the other person actually does. As a result both players accuse each other and a sub-optimal equilibrium is achieved for both players (they both receive 2 year jail terms rather than 1 year terms if they had both remained silent). This is clear from the expected utilities, as row increases Q

_{r}this in turn increases row’s negative contribution to col’s expected utility by a factor of two, likewise as col increases Q

_{c}col’s negative contribution to the expected utility of row increases by a factor of two. The net effect is that each player drives the other to the position of mutual accusation because that is the direction of the utility gradient for changes in their respective behavioural strategies. Note that this path is not itself a path through equilibrium solutions, the only equilibrium is the Nash equilibrium at the end of this path where Q

_{x}=1 (p

_{x}(accuse) = 1) for both players.

_{x}: Q

_{r}= tanh[β

_{r}] and Q

_{c}= tanh[β

_{c}]. Substituting these values into the expected utilities and differentiating with respect to β

_{x}:

_{x}terms and will ultimately arrive at the original Nash Equilibrium of Q

_{x}=1, except that now the path they traverse in arriving at the point is a path through equilibrium solutions to the QRE. In this game the QRE is little different in practical terms from conventional game theory except that sub-rational equilibria, i.e., noisy and not perfectly optimising, can be described as a continuous, smooth surface parameterised by a psychological variable. It is not always so simple though as shown in the next example.

#### 3.2. The QRE and the non-linear effects of group interactions

_{r}= β

_{c}≡ β and the utility matrices (1): U

_{r}≡ transpose[U

_{c}]. Now instead of having two different types of agents in the system each with their own β

_{r}and β

_{c}values and their own individual utilities we have only one type with discriminability β and universal incentives. Then the identity Q

_{r}= Q

_{c}≡ Q holds and:

_{x}(U

_{x}) of Equation (32). The red and blue contours illustrate the mean field equivalent of the principal branches (see Section 4) while the black contour shows a pitchfork bifurcation that is known to be a second order phase transition [39]. The right plot illustrates variation in the β parameter for three different values of the social field h. For these different values the system transitions from low h to high h (i.e., constant monotonic increase) in three distinctive ways; smoothly (red curve), rapidly but still maintaining contact with the equilibrium surface (black curve) or discontinuously where the system suddenly jumps from one surface to the other as h smoothly increases (blue curve).

**A key aspect**of the QRE though is that there are a number of important macroscopic properties of the QRE surface, notably the appearance of unavoidable tipping points and the dangerous erosion of equilibrium islands (also described in the next section) that are lost if we were to use Equation (29) as an approximation to a non-homogeneous population for which the QRE of Equations (22)–(23) are the equilibrium solutions. Some of these properties have only recently been explored [10] and so the following section introduces some of these so far unexplored properties as a function of the underlying system parameters.

## 4. Quantal Equilibrium Paths: Perturbed Cooperation and the principal branch of the QRE

_{r}= δ

_{c}=0 recovers E(U

_{x})= Q

_{c}Q

_{r}. The QRE fixed points change as a function of the β

_{x}terms, plots of these equilibrium surfaces in terms of β

_{r}and β

_{c}will be used to qualitatively illustrate significant outcomes in social interactions. Substituting the utilities from table (33) into Equations (6)–(9) we find: ${f}_{x}({U}_{x})={\scriptstyle \frac{1}{4}}{\delta}_{x}$

_{row}=0 where there is only a single fixed point (β

_{col}= constant) to the perfect rationality solution as β

_{row}→∞ (alternatively β

_{col}→∞ and β

_{row}= constant), the existence of a principal branch is proven in McKelvey and Palfrey [17] based on the earlier work of Harsanyi [45] and the Nash Equilibrium achieved as both β

_{row}and β

_{col}→∞ is called the limiting logit equilibrium of the game.

_{r}and δ

_{c}terms in different ways the response of the system to mild parametric perturbations can be studied. Figure 4 shows the positive β

_{x}quadrant for a range of values of δ

_{r}= δ

_{c}where the distinctive separation (bifurcation) of equilibrium surfaces occurs in a generalised fashion (the left and right plots of Figure 4) to that which is studied in mean-field models of social interactions [6].

#### 4.1. The stranding of sub-optimal strategies on equilibrium islands

_{r}and Q

_{c}for the following three cases of the QRE based on tables (33):

_{x}terms can be interpreted as a mutual deterioration of the joint strategy {B, R} in table (33) as both δ values decrease from positive to negative values. To continue the economic example of fuel for cars, this can be thought of as both petrol powered cars and petrol as a fuel becoming more expensive relative to the alternative of electric powered cars and supplying consumers electricity at outlets for their cars. As petrol prices at gas stations increase and manufacturing costs increase (for example through increased shipping and transport costs of raw materials due to more expensive petrol and diesel), both manufacturing petrol powered cars and buying petrol to power those cars becomes more expensive, thereby decreasing the utility of this solution to both market sectors.

_{row}and β

_{col}. As δ decreases the previously good strategy is now isolated from the origin (in the right plot) and so is no longer on the principal branch for either player and the expected utility in the lower plot is no longer optimal for this β pair.

#### 4.2. The instability of fixed strategies located on equilibrium islands

_{r}= δ

_{c}perturbations to the utilities continue to decrease. If the original β pair remains fixed at β

_{r}= β

_{c}=2 the bifurcation region expands towards the given strategy and eventually overtakes it, forcing the economy through a tipping point even though the β values remain fixed.

_{r}= δ

_{c}→ 0 the gap between the two surfaces decreases, eventually reducing to zero when δ

_{r}= δ

_{c}=0. If the direction of change in these δ terms is reversed the opposite effect is observed, the gap between the surfaces increases and consequently the tipping point becomes more significant the further the chosen strategy deteriorates.

#### 4.3. Intermingling of optimal and sub-optimal strategies via selective incentive variation

_{r}= β

_{c}≃ 2 and Q

_{row}= Q

_{col}≃ 1 is the optimal (for the given β pair) for row but it is sub-optimal for col. This is shown in the bottom row of plots of the two expected utilities.

_{col}is also a tipping point in Q

_{row}for some continuous variation in the β pairs. A consequence of this is the inter-folded aspect of the utility surfaces, E(u

_{col}) shows an intersection between the equilibrium utility surfaces along the β

_{c}axis whereas the E(u

_{row}) shows an intersection in the utility surfaces along the β

_{r}axis. The intersection of course represents equivalent utility values where the corresponding Q values are very far apart in the strategy space, a surprising and perhaps even a counterintuitive result of the QRE.

_{c}=2 and allowing β

_{r}to vary smoothly from 0 to 2 the principal branch of row results in Q

_{col}≃ −1 in the β

_{r}→ ∞ limit. Alternatively, setting β

_{r}=2 and allowing β

_{c}to vary smoothly from 0 to 2 the principal branch of row results in Q

_{row}≃ +1 in the β

_{r}→ ∞ limit, clearly the principal branch of one player does not correspond to the principal branch of the other player in the large β limit. This necessitates the inter-folded nature of the utility surfaces discussed above, fixing β

_{r}=2 and allowing β

_{c}to vary from 0 to 2 we see in the top two plots of Figure 6 there is a smooth path in the equilibrium surfaces for both Q

_{col}and Q

_{row}terminating at β

_{r}= β

_{c}=2.

_{r}= β

_{c}→ 0 then what necessary changes to the incentive structure would sufficiently distort the topology of the QRE such that a tipping point could be averted? As shown in Figure 7 the incentives can readily change the location of the tipping points such that a constant and equal variation in the β parameters would not result in the system passing through a tipping point.

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Plot of the QRE fixed point solutions of the Prisoner’s Dilemma game (matrices (24)). (

**Left**): Expected utility to Col, Equation (25); (

**Right**): Expected utility to Row, Equation (27). The QRE for high β values are shown with black points, as each β → ∞ this becomes a better approximation to the Nash Equilibrium of the game.

**Figure 2.**Plot of fixed point solutions of Equation (29); (

**Left**): mean field contours for fixed h values; (

**Right**): mean field contours for fixed β values.

**Figure 3.**The tracing procedure and tipping points illustrated. The left figure is the QRE surface as given by Equation (22) and the right figure is the expected utility for each {Q

_{r},Q

_{c}}equilibrium pair for an example 2 × 2 game. The black paths illustrate the tracing procedure: beginning at β

_{row}near 0, there is a unique path called the principal branch continuously connecting the origin to an equilibrium point as β

_{row}→ ∞. A tipping point (illustrated with the red line) occurs when the state of the system is changes (here from Q

_{row}≃ 1) smoothly as an underlying system parameter varies (here β

_{row}≃ 2 decreasing to β

_{row}≃ 0.6), at some point along this path an abrupt change occurs when a further small change in β

_{row}results in a large change in Q

_{row}as the system no longer has a local equilibrium and transitions (out of equilibrium) to another equilibrium point, the β value at which this abrupt transition occurs is a tipping point.

**Figure 4.**Perturbed QRE solutions for δ

_{c}= δ

_{r}∈{0.2, 0, −0.2} from left to right with a β pair β

_{c}= β

_{r}=2, the equilibrium strategy is where the black dot is, see Equations (38)–(39).

**Figure 5.**The game as in table (33) with δ

_{r}= δ

_{c}∈ {−2, −4, −6} from left to right in the figure. As the incentives for the original (globally optimal) solution decrease with β

_{r}and β

_{c}fixed the probabilities represented by Q

_{col}(shown, top row) and Q

_{row}(not shown) slowly change but the “equilibrium island” is decreasing in size. The expected utility (bottom row) of the equilibrium point decreases slowly while the viable equilibrium region contracts and leads to only a single viable equilibrium point that can only be reached by a drastic (non-equilibrium) shift in strategy.

**Figure 6.**Perturbed QRE solutions for both players for Q values and the corresponding expected utilities: {δ

_{r},δ

_{c}} = {0.2, −0.2}.

**Figure 7.**Perturbed QRE solutions for (from left to right in the figure): {δ

_{r},δ

_{c}} = {0.16, −0.24}, {0.2, −0.2}, {0.24, −0.16}. As β

_{r}= β

_{c}→ 0 the left figure has a tipping point, the centre figure has a continuous (but not smooth) bifurcation and the right plot has no tipping point, see [10]. Note that the front vertical plotting frame of the figure is aligned with the point of view, decreasing the β terms is equivalent to the equilibrium black dot moving away from the viewer along the line of the front vertical frame. So for β

_{r}= β

_{c}→ 0 the equilibrium solution can change smoothly (right plot) or passes through a tipping point (left plot) depending on the type of perturbation applied to the underlying utility structure.

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Harré, M.S.; Bossomaier, T. Strategic Islands in Economic Games: Isolating Economies From Better Outcomes. *Entropy* **2014**, *16*, 5102-5121.
https://doi.org/10.3390/e16095102

**AMA Style**

Harré MS, Bossomaier T. Strategic Islands in Economic Games: Isolating Economies From Better Outcomes. *Entropy*. 2014; 16(9):5102-5121.
https://doi.org/10.3390/e16095102

**Chicago/Turabian Style**

Harré, Michael S., and Terry Bossomaier. 2014. "Strategic Islands in Economic Games: Isolating Economies From Better Outcomes" *Entropy* 16, no. 9: 5102-5121.
https://doi.org/10.3390/e16095102