6SSPP403 Advanced Game Theory Project

Candidate Number 1- AE30037

Candidate Number 2- AE06015

Word Count- 1570

Title: Strategic Allocation in the Zero-Sum Game of Territorial Expansion

Introduction

In the ever-evolving discourse of political economy, the strategic interactions between states
or groups frequently mirror the complex, competitive scenarios envisaged in game theory.
This essay introduces a novel game framework, termed the "Territorial Expansion Game,"
which is designed to explore the strategic dynamics of resource allocation and territorial
control among competing entities. This game aims to elucidate the normative and
behavioural solutions to conflicts arising in a zero-sum setting where the gain of one
participant directly corresponds to the loss of another, using a repeated assurance game
modelled in a Bayesian format.

Literature Review

In the specialised realm of game theory applied to international relations, particularly in the
context of territorial disputes, there has been significant evolution from the classical
concepts developed by Von Neumann and Morgenstern. Scholars like Powell (2006) have
utilised non-cooperative game theory to explore conditions that may lead states to choose
war or peace over disputed territories (Powell, 2006). This approach typically employs zero-
sum games to model scenarios where one state's gain is another's loss, a principle central to
the Territorial Expansion Game.

Further advancements in the field include the use of Bayesian games, which introduce
uncertainty into the decision-making processes of states. The works of Fey and Ramsay
(2010) incorporate elements of incomplete information, requiring states to make strategic
decisions without full knowledge of others' intentions (Fey & Ramsay, 2010). This
methodology reflects the dynamics of the Territorial Expansion Game, where states update
their beliefs based on prior interactions and signals, thus influencing their strategies on
whether to pursue territorial expansion.

Moreover, the exploration of repeated games, emphasised by Axelrod (1984), focuses on

long-term interactions and strategies, illuminating how consistent engagements and
ongoing relationships influence the strategic choices of states (Axelrod, 1984). In these
games, the history of play can enforce cooperation or prompt retaliation, providing a richer
framework for understanding the dynamics at play in the Territorial Expansion Game.
Decisions are not isolated but are influenced by past actions and anticipated future
interactions, thereby affecting the overall stability and outcomes of regional politics.

Overall, the robust theoretical foundation at the intersection of game theory and
international relations not only enriches academic discussion but also enhances our
understanding of conflict resolution and strategic decision-making in geopolitics. By
integrating empirical research from political science and economics, the Territorial
Expansion Game emerges as a critical analytical tool for understanding and strategising in
territorial disputes.

Game Design
The Territorial Expansion Game is conceptualised as a zero-sum, repeated assurance game
involving three sovereign entities—States A, B, and C—competing for control over a disputed
territory. The game is structured as follows:

 Players: States A, B, and C

 Strategies: Each state can choose either to "Expand" or "Refrain" from territorial
expansion.
 Payoffs: The payoff for each state is contingent upon the combination of strategies
chosen by all players. If a state chooses to expand while the others refrain, it gains
significant territorial control, represented quantitatively. If more than one state
chooses to expand, all involved incur costs due to conflict, reducing their payoffs
substantially. If all refrain, the status quo remains with minimal gains.

The Bayesian format introduces uncertainty by assuming that each state has incomplete
information about the others' willingness to expand. Each state estimates probabilities
regarding the others' decisions based on historical interactions and current political climates.

Territorial Expansion Game: Payoff Calculations and Analysis

To deepen the understanding of the strategic dynamics in the Territorial Expansion Game, let
us examine the payoffs and the resulting calculations in the context of Nash equilibrium.

Payoff Matrix

The payoff for each state varies based on their choice and the choices of others:

 If a state expands while others refrain, it gains significant territorial advantage (e.g.,
+100 units).
 If a state expands and at least one other state also expands, all expanding states
incur a conflict cost (e.g., -50 units each).
 If all states refrain, there is no change in territorial control (0 units).

Payoff Conditions:

RRR (All Refrain): 0 for all players, as no state gains or loses territory.

ERR, RER, RRE (One Expands): +100 for the expander, -50 for the refrainers.

EER, ERE, REE (Two Expand): -10 for expanders due to conflict costs, 0 for the refrainer.

EEE (All Expand): -30 for each due to escalated conflicts, reflecting high costs.

State A\State RR RE ER EE
B,C
E 100,-50,-50 -10,0,-10 -10,-10,0 -30,-30,-30
R 0,0,0 -50,-50,100 -50,100,-50 0,-10,-10
Expected Utility Calculations

Each state must consider its strategy's expected utility, contingent on its beliefs about the
others' strategies. Assume each state initially believes there is a 50% chance that any other
state will expand.

Expected Utility for State A:

State A Expands (E):

Probability of B and C both refraining (R,R): 0.5 x 0.5 = 0.25

Probability of only B expanding (E,R): 0.5 x 0.5 = 0.25

Probability of only C expanding (R,E): 0.5 x 0.5 = 0.25

Probability of both B and C expanding (E,E): 0.5 x 0.5 = 0.25

Expected Utility (EU_E^A) = (0.25 * 100) + (0.25 * -10) + (0.25 * -10) + (0.25 * -30) = 25 - 2.5 -
2.5 - 7.5 = 12.5

State A Refrains (R):

Expected Utility (EU_R^A) = (0.25*0) + (0.25*-50) + (0.25*-50) + (0.25*0) = 0 -12.25 – 12.25

+ 0 = -25

Calculations for States B and C:

Following the same structure as for State A, using symmetrical beliefs:

State B Expands (E):

Expected Utility (EU_E^B) calculated similarly to A, as the game is symmetric = 12.5

State B Refrains (R):

Expected Utility (EU_R^B) = -25 (Similar to A, as game is symmetric)

State C Expands (E):

Expected Utility (EU_E^C) calculated similarly to A, as the game is symmetric = 12.5

State C Refrains (R):

Expected Utility (EU_R^C) = -25 (Similar to A, as game is symmetric)

Calculating the Mixed Strategy Nash Equilibrium

We seek to calculate whether there exists a Nash equilibrium in mixed strategies where each
state's decision to expand or refrain is a best response to the others' mixed strategies.

Each state compares the expected utility of expanding versus refraining. Given our
calculations:

For State A: EU_E^A (12.5) > EU_R^A (-25)

For State B: EU_E^B (12.5) > EU_R^B (-25)

For State C: EU_E^C (12.5)> EU_R^C (-25)

Under these initial beliefs (50% chance of expansion by any state), the strategy of expanding
has a higher expected utility for each state, indicating a tendency towards expansion.
However, this scenario leads to mutual detriment (All Expand), suggesting a misalignment
between the Mixed Strategy Nash equilibrium and the optimal collective outcome. The
calculations reveal a Mixed Strategy Nash equilibrium where each state chooses to expand,
driven by the individual rationality of maximising expected utility under the given
probabilities. However, this equilibrium is not Pareto optimal, as it results in a higher cost
due to conflict. The game illustrates the classic security dilemma where states, acting based
on incomplete information and individual rational calculations, might end up in a suboptimal
Pareto inefficient outcome; All individuals would prefer to RRR(0) to EEE(-30).

Experimenting with different variations of the game

The assumption that each state has a 50% chance of expanding isn’t realistic, hence we
could assign individual probabilities based on their aims/preferences as a state.

 State A has a 90% chance of expanding given their pursuit for world domination
 State B has a 30% chance of expanding due to their passive nature and inclination
towards anti-aggression policies
 State C has a 50% chance of expanding based off of their partially conflicting aims of
gaining power and peace.

Moreover, the payoff, for RRR can be represented by all states gaining +50 as they enjoy the
economic benefits during peaceful times where costs are low.

Similarly, the state that chooses to refrain while the others expand can also enjoy a payoff of
+50 as they don’t incur any costs, and relish in the economic benefits of investing in sectors
other than military/defence.

Additionally, the costs of two states expanding can also be increased to -30, reflecting the
huge loss of human lives during war.

Adjusted Payoff Conditions:

RRR (All Refrain): =50 for all players, as +50 as they enjoy the economic benefits during
peaceful times, no costs for defence.

ERR, RER, RRE (One Expands): +100 for the expander, -50 for the refrainers.

EER, ERE, REE (Two Expand): -30 for expanders due to conflict costs/lives lost, +50 for the
refrainer.

EEE (All Expand): -30 for each due to escalated conflicts, reflecting high costs.

Adjusted Payoff Matrix:

State A\State RR RE ER EE
B,C
E 100,-50,-50 -30,50,-30 -30,-30,50 -30,-30,-30
R 50,50,50 -50,-50,100 -50,100,-50 50,-30,-30

Adjusted Expected Utility Calculations:

Expected Utility for State A:

State A Expands (E):

Probability of B and C both refraining (R,R): 0.7 x 0.5 = 0.35

Probability of only B expanding (E,R): 0.3 x 0.5 = 0.15

Probability of only C expanding (R,E): 0.7x 0.5 = 0.35

Probability of both B and C expanding (E,E): 0.3 x 0.5 = 0.15

Expected Utility (EU_E^A) = (0.35 * 100) + (0.15 * -30) + (0.35 * -30) + (0.15 * -30) = 35 – 4.5
– 10.5 – 4.5 = 15.5

State A Refrains (R):

Expected Utility (EU_R^A) = (0.35*50) + (0.15*-50) + (0.35*-50) + (0.15*50) = 10.5 – 7.5 –

10.5 + 7.5 = 0

State B’s Expected Utility Calculation

Given:

• Probability of State A expanding (E): 90%

• Probability of State C expanding (E): 50%

State B Expands (E):

• Probability of A and C both refraining (R,R): 0.1 x 0.5 = 0.05

 • Probability of only A expanding (E,R): 0.9 x 0.5 = 0.45

• Probability of only C expanding (R,E): 0.1 x 0.5 = 0.05

• Probability of both A and C expanding (E,E): 0.9 x 0.5 = 0.45

Expected Utility (EU_E^B) = (0.05 * 100) + (0.45 * -30) + (0.05 * -30) + (0.45 * -30) = 5 – 13.5
– 1.5 – 13.5 = -23.5

State B Refrains (R):

Expected Utility (EU_R^B) = (0.05 * 50) + (0.45 * -50) + (0.05 * -50) + (0.45 * 50) = 2.5 -22.5 -
2.5 + 22.5 = 0

State C’s Expected Utility Calculation

Given:

• Probability of State A expanding (E): 90%

• Probability of State B expanding (E): 30%

State C Expands (E):

• Probability of A and B both refraining (R,R): 0.1 x 0.7 = 0.07

• Probability of only A expanding (E,R): 0.9 x 0.7 = 0.63

• Probability of only B expanding (R,E): 0.1 x 0.3 = 0.03

• Probability of both A and B expanding (E,E): 0.9 x 0.3 = 0.27

Expected Utility (EU_E^C) = (0.07 * 100) + (0.63 * -30) + (0.03 * -30) + (0.27 * -30) = 7 – 18.9
– 0.9 – 8.1 = -20.9

State C Refrains (R):

Expected Utility (EU_R^C) = (0.07 * 50) + (0.63 * -50) + (0.03 * -50) + (0.27 * 50) = 3.5 – 31.5
– 1.5 + 13.5 = -16

For State A: EU_E^A (15.5) > EU_R^A (0)

For State B: EU_E^B (-23.5) < EU_R^B (0)

For State C: EU_E^C (-20.9)< EU_R^C (-16)

Adjusted Strategic Outcomes and Nash Equilibrium

**Revised Analysis of Strategic Outcomes and Nash Equilibrium**

In the adjusted Territorial Expansion Game, the unique strategic inclinations of States A, B,
and C significantly influence their optimal strategies, affecting the Nash Equilibrium. State A,
with a 90% chance of expansion, continues to favour aggressive expansion, as evidenced by
its much higher expected utility for expanding (EU_E^A = 15.5) compared to refraining
(EU_R^A = 0). This aggressive posture solidifies expansion as State A's dominant strategy.

In contrast, States B and C shift towards more defensive strategies due to their respective
passive and conflicted natures, combined with the higher conflict costs and the significant
threats posed by State A's aggressiveness:

 State B finds refraining more advantageous (EU_R^B = 0) over expanding (EU_E^B = -

23.5) reflecting its preference to avoid the heightened risks of conflict.
 State C also opts to refrain, as the lesser negative utility of refraining (EU_R^C = -16)
compared to expanding (EU_E^C = -20.9) indicates a strategic withdrawal in the face
of potential losses.

Thus, the mixed-strategy Nash Equilibrium results in State A expanding, driven by its
aggressive strategy, while States B and C refrain, opting for the safer route to mitigate risks
and avoid the escalated costs of potential conflicts. This strategic divergence underscores
the complexities and interdependencies in international relations, where the aggressive
policies of one state can dictate defensive responses from others, illustrating a delicate
balance of power within the game's framework.

Limitations

The Territorial Expansion Game, while insightful, has notable limitations. It assumes all
actors are rational and solely focused on utility maximisation, which oversimplifies the
complexities of international relations that often involve emotional, historical, or other non-
rational factors. The model also fails to consider external influences such as international
law, global public opinion, and supranational organisations, which could limit its real-world
applicability. Additionally, it treats states as monolithic entities, overlooking internal political
dynamics and the influence of domestic stakeholders. Addressing these limitations could
enhance the game’s relevance and predictive power in understanding geopolitical strategies.

Conclusion

The Territorial Expansion Game provides a structured way to analyse and predict the
outcomes of strategic interactions in a political economy context, especially under
conditions of uncertainty and incomplete information. This game theoretical model not only
highlights the potential for conflict and cooperation but also stresses the importance of
understanding the interdependencies and information asymmetries that influence decision-
making processes. Through this framework, the nuanced complexities of real-world
geopolitical scenarios can be better understood, offering insights into both the causes of
conflicts and the possibilities for peaceful resolutions. This model thus contributes to the
broader understanding of strategic interactions in political economy, enhancing our ability to
apply game theory to complex, real-world problems.

Bibliography
Powell, R., 2006. War as a Commitment Problem.. International
Organization, 60(1), pp. 169-203.

Mark Fey, K. W. R., 2011. Uncertainty and Incentives in Crisis Bargaining:

Game-Free Analysis of International Conflict.. American Journal of Political
Science, 55(1), pp. 149-69.

Axelrod, R. (1984). The Evolution of Cooperation. New York: Basic Books.