# Modelling the Dynamics of Identity and Fairness in Ultimatum Game

Janvi Chhabra
International Institute of Information Technology, Bangalore
26/C, Electronics City Phase 1
Bangalore, Karnataka, India
<EMAIL_ADDRESS>
&Jayati Deshmukh
International Institute of Information Technology, Bangalore
26/C, Electronics City Phase 1
Bangalore, Karnataka, India
<EMAIL_ADDRESS>
&Srinath Srinivasa
International Institute of Information Technology, Bangalore
26/C, Electronics City Phase 1
Bangalore, Karnataka, India
<EMAIL_ADDRESS>

###### Abstract

Allocation games are zero-sum games that model the distribution of resources
among multiple agents. In this paper, we explore the interplay between an
elastic sense of subjective identity and its impact on notions of fairness in
allocation. An elastic sense of identity in agents is known to lead to
responsible decision-making in non-cooperative, non-zero-sum games like
Prisoners’ Dilemma, and is a desirable feature to add into agent models.
However, when it comes to allocation, an elastic sense of identity can be
shown to exacerbate inequities in allocation, giving no rational incentive for
agents to act fairly towards one another. This lead us to introduce a sense of
fairness as an innate characteristic of autonomous agency. For this, we
implement the well-known Ultimatum Game between two agents, where their
elastic sense of self (controlled by a parameter called $\gamma$) and a sense
of fairness (controlled by a parameter called $\tau$) are both varied. We
study the points at which agents find it no longer rational to identify with
the other agent, and uphold their sense of fairness, and vice versa. Such a
study also helps us discern the subtle difference between responsibility and
fairness when it comes to autonomous agency.

_Keywords_ Identity, Allocation, Allocation game, Responsibility, Fairness

##  1 Introduction

The allocation of limited resources amongst individuals or groups with
competing needs often creates a dilemma especially when conflicting goals,
interests, or values are involved. This can be classified under a broad
umbrella of responsibility dilemma, where an agent faces a conflict between an
individually optimal state and a collectively or socially optimal state. These
dilemmas are relevant in the context of both individuals in human societies as
well as autonomous agents in multi-agent systems [7].

In such dilemmas, the classical game theory models agents to act rationally
i.e. they are modelled to choose actions which maximise their payoff. However,
often human societies don’t demonstrate such selfish behaviour; humans in such
dilemmas do consider factors more than just their personal benefit [23].
Responsible behaviour often emerges in human populations even in times of
extreme conflict and oppression [3]. Such behaviour indicates that agents’
decision-making should be more nuanced, maximising not just their own payoff
but also being aware of the consequences of their actions on others.

The notion of fairness in agents has been widely studied in resource
allocation scenarios [1]. The notion of fairness of an agent attributes the
allocation which an agent perceives as fair.

Humans facing the dilemma in allocation games demonstrate a preference towards
fairness in addition to personal benefit [9]. They perceive the allocation
through the lens of their notion of fairness and hence the utility function
which they try to maximise is also influenced by their fairness preference.

The notion of fairness in agents has been modelled in diverse ways. In some
models, agents’ notion of fairness is modelled as conforming with norms such
as inequity aversion and reciprocity, where adhering to these norms is
incorporated into the agents’ utility function [12]. Another way the notion of
fairness has been approached is by considering the influence of emotion and
cognition in agents’ utility functions [24]. Reinforcement learning based
approaches have also been used by agents to learn human strategies in the
scenario of an ultimatum game.

We extend a recently proposed model of responsible agency called Computational
Transcendence (CT) which models an elastic identity in autonomous agents and
it has been demonstrated to result in emergent responsible behaviour [10]. We
introduce a notion of fairness into this model and then agents using this
extended model are evaluated in the context of allocation games.

In this paper, we specifically look at the Ultimatum Game (UG) as an
allocation scenario. This game highlights the different aspects involved in
the decision-making process such as the notion of fairness in agents. The
decisions of agents in social dilemmas such as UG have been interpreted in
various ways, and in this work, we explore how responsible autonomous agents
with a model of fairness act specifically in the context of the ultimatum
game.

##  2 Related Work

In this section, we discuss in detail the ultimatum game (UG) and elaborate
rational human behaviour in such a scenario. We also describe some existing
approaches to model fairness in the ultimatum game. Further, we discuss a few
frameworks that point to the importance of the subjective notion of fairness
in the agents. Finally, we introduce the existing CT model and introduce
fairness in this model in the context of UG.

###  2.1 Ultimatum Game

Figure 1: The Ultimatum Game

The Ultimatum Game (UG) was introduced by three economists Güth,
Schmittberger, and Schwarze in 1982 [13]. In the standard UG between two
agents, there exist two roles: Allocator and Recipient. For a limited resource
$R$, the allocator proposes the allocation of the resource amongst itself and
the recipient. Then the recipient can either accept or reject the proposed
allocation.

  * •

If the recipient accepts the proposed allocation, both allocator and recipient
receive the split proposed by the allocator.

  * •

If the recipient rejects the proposed allocation, both allocator and recipient
receive nothing.

Figure 1 summarises the ultimatum game between the allocator and the
recipient. This game has led to significant research in analysing the trade-
off between personal and comparative outcomes [18]. It has been introduced to
test the theoretical assumption that utility maximisation can be equated to
maximising personal monetary payoffs [13]. Also, it has given rise to numerous
other games like the dictator game, trust game, gift exchange game, and public
goods game [25].

###  2.2 UG in Humans

If the players are assumed to be rational as per the classical rational
theory, it means that players will optimise to maximise their external payoff.
Following this assumption, as per the game equilibrium, the allocator should
offer the smallest possible split of the resource to the recipient, as it
knows that the recipient can either choose between obtaining the proposed
split by accepting or obtain nothing by rejecting the deal. And it makes
rational sense for the recipient to accept the proposed split as accepting the
deal would maximise its payoff.

However, this rational assumption often is not observed in the case of
experimental studies conducted on humans. The experimental results, when
humans played the ultimatum game as a part of the experimental setup suggest
that the modal offer that the allocator proposes is often around 50$\%$ and
offers $\leq$ 20$\%$ have around 50$\%$ chance of getting rejected [4] by the
recipient. These findings suggest that humans don’t just maximise their
external payoff, but are also concerned about the comparative payoff, which
indicates that they have a notion of fairness. UG became a popular game in
experimental studies to understand the behaviour of allocator and recipient
when humans play these games. A few of the scenarios which were experimentally
tested are as follows:

  * •

When the stake of the resource to be allocated is increased: Even a 20 times
increase in stake didn’t have a significant impact on allocator and recipient
behavior [19].

  * •

When the players were known or anonymous to each other: When the players were
known to each other, the allocator usually offered a greater split to the
recipient as compared to when they were anonymous [2].

  * •

When UG was played across various cultures: It was observed that variation in
culture leads to very diverse behaviour. In most dominant cultures, the game
settles in the state as discussed so far. However, in some cultures, the
settled state of the game is different from usual. For example, in the
Peruvian tribe in the Amazon, the allocator offers a very low share (around
26$\%$) to the recipient and faces a very high rejection rate from recipients
[22]. In contrast to this, in the case of Lamelara tribes of Indonesia, the
allocator offers a very high share (around 58$\%$) to the recipient and yet
approximately 37$\%$ of such deals are rejected by recipients [5].

This evidence from experimental studies suggests that humans in the UG
scenario do not behave purely based on the rational (payoff maximization)
assumption. The evidence, especially from cultural studies indicates that the
comparative payoff is interpreted in diverse ways across cultures. This also
indicates the effect of culture on the identity of humans, which influences
how they perceive such UG scenarios.

###  2.3 Modelling Fairness in UG

Next, we discuss various ways in which the modelling of agents’ behaviour has
been approached specifically in the UG scenario.

####  2.3.1 Inequity Aversion based Model

Fehr and Schmidt [12] proposed an inequity-aversion-based model of fairness in
the UG context. An agent is inequity-averse when it dislikes the inequitable
allocation of resources irrespective of which side of the inequity it lies on.
This inequity-averse nature of agents affects their utility function. They
assume that being on the disadvantageous side of inequity gives a more
negative utility as compared to being on the advantageous side of the
inequity.

####  2.3.2 Reciprocity based Model

Rabin, Falk and Fischbacher proposed a reciprocity-based model of fairness
[21, 11]. Using this model, agents evaluate the “kindness” of the action of
the other agent they are interacting with and respond or reciprocate
accordingly. This evaluation of kindness constitutes factoring in the
consequences of the other agent’s actions and the underlying intentions behind
those actions.

The utility function of the agents is influenced by the kindness of the action
and the reciprocation reaction of agents to that treatment. Agents’ actions
are influenced by their first-order and second-order beliefs. They try to
maximise their utility and align their actions with their beliefs.

####  2.3.3 Emotion and Cognition based Model

Tamarit and Sánchez [24] proposed a model where agents’ behaviour is
influenced by a combination of both their emotions and cognition. It is
inspired by Kanheman’s two-system theory [16], which proposes that the
decision-making process is a result of the interaction between two systems;
System 1 which is fast, intuitive, associative, and emotionally driven and
System 2 which is slow, cognitively demanding, and analytical.

####  2.3.4 Social Decision-Making in Ultimatum Game

The Social Utility framework proposed for UG [14] categorises the factors
which influence the behaviour of agents in the game. These broad
categorisations are as follows:

  * •

Contextual Factors: Contextual factors in UG are changing the nature of the
resource to be allocated, framing the rules of the game, and the game being
played across diverse sets of people.

  * •

Characteristics of Players: This can be attributed to the social distance
between the players i.e. whether they are known or anonymous to each other.
This can also account for the power dynamics between the players and the
personality of the players which can influence the motivation of the players.

  * •

Characteristics of the Game: This is attributed to the way in which the game
is structured. Different variants of UG, such as the dictator game can also
give an idea about the factors of the game itself which impact certain
behaviour of players.

###  2.4 Computational Transcendence

Computational Transcendence (CT) [10] models an elastic identity or a sense of
self in autonomous agents using which they can identify with external entities
like other agents, groups and notions in the system. The sense of self of an
agent $a$ is represented as $S(a)=(I_{a},d_{a},\gamma_{a})$ where $I_{a}$
represents the identity set of the agent consisting of aspects it identifies
with, $d_{a}$ is the semantic distance of the agent which denotes the
perceived logical distance of an agent to each aspect in its identity set and
$\gamma_{a}$ is the transcendence level of the agent which denotes the extent
to which it identifies with others. An agent $a$, with transcendence level
$\gamma_{a}$ identifies with an aspect $o$ whose distance is $d_{a}(o)$ with
an attenuation factor of $\gamma_{a}^{d_{a}(o)}$. It has been shown that this
model leads to emergent responsible behaviour by autonomous agents across
diverse scenarios. In this paper, first, we test transcended agents in the UG
setup and then extend the model by incorporating a notion of fairness in
transcended agents.

Identifying with external entities affects how an agent’s internal valuation
or utility is computed based on external rewards or payoffs that may be
received by different aspects in its identity set. For any aspect $o\in
I_{a}$, let the term $\pi_{i}(o)$ refer to the payoff obtained by aspect $o$
in the game or system state $i$. Given this, the utility derived by agent $a$
in system state $i$ is computed as follows:

$\begin{split}u_{i}(a)&=\frac{1}{\sum_{\forall o\in
I}\gamma_{a}^{d_{a}(o)}}\displaystyle\sum_{\forall o\in
I}\gamma_{a}^{d_{a}(o)}\pi_{i}(o)\end{split}$ (1)

##  3 Baseline Model

Consider the scenario, where the resource $R$ is assigned the value $1$, and
it is to be divided among two agents with the roles of Allocator and
Recipient. The allocator proposes the allocation of the resource valued $1$ as
$x$ for itself and $1-x$ for the recipient. Next, the recipient can either
accept or reject the proposed allocation. If the recipient accepts this
allocation, both allocator and recipient receive the resource (referred to as
payoff) of $x$ and $1-x$ respectively. On the other hand, if the recipient
rejects the proposed allocation, then both of them get nothing i.e. the payoff
for both agents is $0$.

As discussed, the rational strategy using payoff maximization for the
allocator is to propose a minimum possible split of resource $R$ to the
recipient, and the recipient accepts it because rejecting the allocation
results in a payoff of $0$.

Transcended agents make the decision based on the expected utility of their
choices. Suppose the two transcended agents playing UG are represented as
follows: a transcended agent $a_{1}$ having a transcendence level of $\gamma$
and semantic distance $d$ with the other agent $a_{2}$. The utility of
$a_{1}$, when it receives the split of $x$ and $a_{2}$ receives the split of
$1-x$, is computed in Equation 2 as follows:

$util(a_{1})=\frac{x+\gamma^{d}(1-x)}{1+\gamma^{d}}$ (2)

In this scenario,

  * •

If $a_{1}$ is the allocator: It proposes the maximum utility split, i.e. it
computes the utility for all the possible splits and proposes the split that
gives it the maximum utility.

  * •

If $a_{1}$ is the recipient: It accepts if the proposed split gives $\geq$ its
minimum acceptable utility i.e. it calculates the utility of the proposed
split and if it is $\geq$ 0 it accepts, else it rejects the proposal.

Figure 2: Utility of a transcended agent for possible splits for various
$\gamma$ values. The blue circles are Maximum Utility Split and the red
circles are Minimum Acceptable Utility

Figure 2, shows the utility of the transcended agent for all possible splits
when the utility is computed using Equation 2 for various transcendence
($\gamma$) values. We can infer from the plots that the transcended agent as
an allocator takes the whole resource for itself (except at $\gamma=1$) as it
has maximum utility when it allocates the whole resource to itself, as denoted
by the blue circles. Also, as a recipient, the utility accrued is $\geq$ 0, as
observed by red circles in Figure 2, thus it makes sense to always accept the
proposed allocation. Hence, the game state for almost all values of $\gamma$
values involves the allocator taking almost the whole resource (in this case,
the split of 1) and the recipient accepting the proposed allocation. According
to the definition of transcendence, since the agents identify with each other,
they also account for each other’s payoff. However, in the absence of a notion
of fairness, they propose and accept unfair allocations. Next, we introduce a
notion of fairness in transcended agents and evaluate whether it leads to more
diverse and fair allocations.

##  4 Notion of Fairness

The baseline model of transcendence does not result in fair and diverse deals
in the case of allocation games. The more the agents transcend, the more they
identify with the other agent, they find unfair deals acceptable since the
other agent’s utility also contributes to their utility. So we extend this
model by introducing a notion of fairness in agents using a fairness
threshold.

Fairness threshold, ($\tau$) is defined as a threshold below which if an agent
receives an allocation, it perceives it to be unfair. The rationale for agents
having a fairness threshold has been explored in the literature of UG, where
agents reject any offer below their fairness threshold [15, 20, 6]. The
fairness threshold acts as a lens through which the agents perceive the payoff
they receive. The utility computation should account for the perceived payoff
based on the fairness threshold instead of just the payoff received. Thus, the
process of utility computation consists of two stages, first calculating the
perceived payoff for the allocation and then computing the utility based on
the perceived payoff with respect to the fairness threshold.

###  4.1 Perceived Payoff

Figure 3: Perceived Payoff Function

The risk or loss aversion behaviour [17] is usually demonstrated by humans.
For the same value of loss or gain, humans perceive the absolute value of loss
to be greater than the equivalent gain. Risk aversion is relevant, especially
for agents having a notion of fairness and it can be used to compute the
perceived payoff. Every agent has a fairness threshold ($\tau$) and it
perceives the payoff it receives through the lens of its fairness threshold.
An agent $a$ having a fairness threshold of $\tau_{a}$ receives allocation
$x$. Its perceived payoff is computed using a sigmoid function as shown in
Figure 3. The function $f(x-\tau_{a})$, which is a S-shaped function denotes
the loss aversion behaviour of the agents [8]. The disparity between the
received payoff, $x$ and fairness threshold $\tau_{a}$, is $\delta_{a}$. Then
according to the function as shown in Figure 3, we can state that the
inequality $\lvert f(\delta_{a})\rvert<\lvert f(-\delta_{a})\rvert$ holds.

###  4.2 Utility Computation

As discussed, the external payoff agents receive is perceived through a lens
of fairness threshold. Further, the agents compute their utility based on the
perceived payoffs. As the utility computation also involves the payoff of the
aspects the agent identifies with, it is also replaced by the perceived
payoff. Hence, the agent not only perceives its payoff through the lens of its
fairness threshold but also the split of the other agent it identifies with.

Agents perceive the payoff through the lens of their $\tau$, thus this carries
an assumption that other agents’ fairness threshold is also the same as their
fairness threshold. Hence, utility computation is a combination of perceived
payoff (Figure 3) and transcended utility computation (Equation 1). Assuming
that the total resource to be allocated is 1, the utility of the agent $a$,
receiving the split $s$ is described in Equation 3 as follows:

$util(a)=\frac{f(s-\tau)+\gamma^{d}*f((1-s)-\tau)}{1+\gamma^{d}}$ (3)

So far, the utility computation of the agents with a fairness threshold of
$\tau$ was discussed. Next, we explore different possible ways to represent
$\tau$ in agents. The following two alternatives for representing the fairness
threshold in agents will be elaborated and experimental results for both will
be discussed in detail:

  * •

$\tau$ as an agent-based characteristic: In this case, $\tau$ is a
characteristic of the agent itself and it doesn’t vary for different aspects
in the identity set of the agent (similar to transcendence level, $\gamma$).
Thus, no matter how close (low semantic distance) or far (high semantic
distance) the agent is to some aspect in its identity set, its $\tau$ is the
same. This model assumes that the fairness criteria of an agent is universal
and same for everyone and thus it does not vary depending on whom the agent is
interacting with.

  * •

$\tau$ as an association-based characteristic: In this case, the fairness
threshold $\tau$ of an agent corresponds to each aspect in its identity set
and it depends on how the agent associates with that aspect (similar to the
semantic distance, d). In this case, there is a correlation between the
semantic distance and fairness threshold for every aspect. This model assumes
that fairness is not a universal characteristic but rather a notion that
varies on a case-by-case basis depending on the interaction and perception of
individual aspects in the identity set.

##  5 Agent-based Fairness Threshold

In this section, we discuss in detail the decisions that the allocator and
recipient make when they have fixed fairness threshold $\tau$. We also discuss
the effect of varying other agent-specific parameters like $\tau$, $\gamma$,
and $d$. As discussed in Section 3, the allocator proposes a split which gives
it maximum utility and the recipient accepts a split if its utility is $\geq$
its minimum acceptable utility.

###  5.1 Utility Plots

Figure 4: The utility curve of the transcended agents ($\gamma$ = 0.5, $\tau$
= 0.5) w.r.t possible splits while varying the semantic distance of the agent
with another player

Figure 4 shows the variation of the utility of a split for agents with
$\gamma=0.5$ and $\tau=0.5$. These utility curves are plotted for a range of
semantic distances, $d$ of the agent with the other agent with whom it is
playing the UG. The yellow-shaded region in the figure shows the spectrum of
utility for possible splits when $d$ is varied between $0$ to $2.4$.

The red circle shows the splits at which the recipient gets the minimum
acceptable utility and the blue circles show the splits at which the allocator
gets the maximum utility. From this plot, it can be inferred that with the
increase in semantic distance $d$, the split which gives the maximum utility
to the allocator also increases. Hence, as the allocator transcends higher,
its semantic distances reduce and it is more likely to share the resource to a
greater extent with the recipient. On the other hand, as the transcendence
level decreases, the allocator is likely to take the whole resource for
itself.

As the recipient, it is observed that varying the semantic distance does not
have an impact on the acceptable split, as for all values of $d$, the split
that satisfies the minimum acceptable utility is $0.5$. An interesting
observation can be drawn from the nature of the curves with varying semantic
distance. The curve for utility when $d=0$ is symmetric at the split of $0.5$,
and as the distance increases it becomes S-shaped and gets positive utility
from the being on the advantageous side of the split. The utility curve for
$d=0$ shows the characteristic of an inequity-averse agent discussed earlier
as it accrues negative utility being on either side of inequitable outcomes.
As the value of $d$ increases, this nature of inequity-aversion decreases.

Next, to understand the impact of $\tau$ and $\gamma$, we plot similar utility
curves for different combinations of $\tau$ and $\gamma$, as shown in Figure
5. The effect of varying $\tau$ and $\gamma$ on the utility of the agent can
be inferred as follows:

(a) Low $\tau$ and Low $\gamma$

(b) Low $\tau$ and High $\gamma$

(c) High $\tau$ and Low $\gamma$

(d) High $\tau$ and High $\gamma$

Figure 5: Utility curves with different combination for extremes values of
$\tau$ and $\gamma$

  * •

Effect of $\tau$ on utility:
The nature of utility curves when $\tau$ is low is presented in Figure 5(a)
and Figure 5(b). We note that the utility increases till the split of 0.5 and
then it decreases (when $d$ is low), this is because the $\tau$ is 0.2, and
the utility is maximum at 0.5 split of resource because both the agents are
benefiting equally. For any split beyond 0.5, the other agent suffers more,
and as observed for low $d$, the agent shows inequity-averse behaviour, hence,
the utility of the agent decreases. This inequity-averse nature changes as the
semantic distance is increased.

Figure 5(c) and Figure 5(d) show the utility plots when the value of $\tau$ is
high. In the case of low $d$, the utility decreases as the split is increased
till 0.5 split, where the agent has minimum utility and then it increases.
This is because both agents are equally suffering in case of a 0.5 split and
on either side one of them is benefiting. As the value of $d$ is increased,
the agent transcends less, and the utility of the agent increases as the split
is increased.

In the case of low $\tau$, the allocator (represented by the blue circles)
takes splits for itself which are near 0.5 and they increase the split as the
distance with the recipient increases. In the case of the recipient
(represented by the red circles), when the distance is less, they accept a
split of low value, but it increases as the distance increases. Hence, in this
case, many game states exist where the deal between the allocator and
recipient is accepted.

In the case of high $\tau$, an allocator is always motivated to take the whole
split for itself (except $d=0$), (represented by the blue circles). And as a
recipient, the split which gives the minimum acceptable utility is high around
$0.7$ (represented by the red circles). This indicates that in this case, most
of the deals are rejected (except when $d$ of allocator or recipient is 0 from
the other player).

  * •

Effect of $\gamma$ on utility:
When the $\gamma$ of the agent is low, as observed from Figure 5(a) and Figure
5(c), the splits which the allocator takes for itself (represented by the blue
circles) increases rapidly with the increase in distance. Similarly, the
splits at which recipients get the minimum acceptable utility (represented by
the red circles) also increase rapidly compared to an agent with a high value
of $\gamma$.

The effect of $\gamma$ can also be observed in terms of the extent of the
yellow-shaded region. Agents with low $\gamma$ had more area of shaded region
as compared to high $\gamma$ when the value of $\tau$ is kept constant. This
shows that for the agents with low $\gamma$, the nature of the utility curve
changes rapidly with varying $d$ as compared to agents with high $\gamma$ as
shown in Figure 5(b) and Figure 5(d).

###  5.2 Acceptable Splits for the Recipient

(a) Low $\tau$

(b) Moderate $\tau$

(c) High $\tau$

Figure 6: Acceptable Split Matrix with varying semantic distance of recipient
with allocator (recipient with $\gamma=0.4$)

So far, we discussed the utility plots and the variation in the decisions of
the allocator and recipient with the change in $\gamma$, $d$, and $\tau$.
Next, we evaluate the game setup specifically focusing on the conditions which
lead to deals being accepted. We analyse the game from the recipient’s
perspective and study the factors which affect the acceptance or rejection of
the proposed deal. As noted earlier, $\gamma$ does not have a significant
impact on the nature of the utility curve, so we choose a fixed value of
$\gamma$ ($0.4$ in this case). We vary the semantic distance $d$ of the
recipient from the allocator and analyse its impact on the acceptance or
rejection of the split offered to the recipient.

The matrices in Figure 6 represent the split on the x-axis and the variation
in the semantic distance of the recipient with the allocator on the y-axis,
for different values of $\tau$. The acceptance of a split is represented by
green colour and rejected splits by red colour. The following inferences can
be drawn from the matrices:

  * •

Recipient with low $\tau$: In Figure 6(a), when the $\tau$ of the agent is low
($\tau$=0.2), it accepts all the deals where the recipient and allocator are
getting $\geq$ 0.2 up to a certain distance. After that recipient starts
accepting deals in which the allocator is getting less than 0.2.

  * •

Recipient with moderate $\tau$: For the agents with moderate $\tau$
($\tau$=0.5), as shown in Figure 6(b), when the semantic distance of the
recipient with the allocator is very low, it only accepts the 0.5 splits, as
either way getting more or less than that is unfair for one of them. As the
semantic distance with the allocator increases, it starts accepting all the
splits more than 0.5.

  * •

Recipient with high $\tau$: As observed in the above cases, recipients reject
the higher split offers when the semantic distance with the allocator is less.
This is because they also account for the fairness of the allocation of the
allocator. Hence, when the agent’s $\tau$ is high ($\tau$ = 0.7), and the
semantic distance is more, they accept the extreme splits, where at least one
of them is satisfied. As the distance increases, they accept the splits which
are strictly greater than 0.7.

In this section, the fairness threshold ($\tau$) is modelled as a fixed
characteristic of the agent. The agents perceive the allocations through the
lens of this threshold (factoring in their share as well as other player’s
share). As observed in Figure 6(c), most of the deals are rejected when the
agent’s $\tau$ is high, even though the extent to which the agent transcends
($\gamma^{d}$) is high. Hence, in this model, we observe a contradiction in
having a high fairness threshold as the innate characteristic and also highly
transcending with the other agent. For example, a realistic scenario of
resource allocation with high transcendence is sharing food with a family
member. With a higher transcendence level, there is a higher tendency to share
since the agent prefers to a higher extent the well-being of others. In this
case, it doesn’t make sense to have a fairness threshold of $0.7$ (i.e. taking
atleast 70$\%$ of the food for oneself). In this example, having a static
$\tau$ makes sense only in exceptional cases like when health requirements
mandate having a minimum of 70$\%$ of the food. Next, we explore an alternate
formulation where the fairness threshold ($\tau$) of an agent depends on the
association of the agent with the other agent it interacts with.

##  6 Association-based Fairness Threshold

Figure 7: Association-based fairness threshold curves on varying semantic
distance with the identity elements for different $\gamma$ values

(a) Low $\gamma$

(b) Moderate $\gamma$

(c) High $\gamma$

Figure 8: Utility plots for the agents with different values of $\gamma$

(a) Low $\gamma$

(b) Moderate $\gamma$

(c) High $\gamma$

Figure 9: Acceptable Split Matrix with varying semantic distance of recipient
with allocator for different values of $\gamma$

In this section, we introduce the notion of fairness using an association-
based fairness threshold for the agents. It is similar to the semantic
distance, d that the agents have with different aspects in their identity set.
Agent a, has a fairness threshold $\tau$, for each element o in its identity
set represented as $\tau_{a}(o)$.

As discussed earlier, transcended agents account for the payoff of the aspects
they identify with in their utility computation as described in Equation 1.
Their fairness threshold should thus correlate with the extent of their
transcendence level towards the other agent they are interacting with. The
extent of transcendence of an agent with transcendence level $\gamma$ and
semantic distance $d$ with the other agent is denoted using $\gamma^{d}$.
Using this formulation, we define association-based fairness threshold
($\tau_{a}(o)$) towards aspect o in Equation 4 as follows:

$\tau_{a}(o)=1-\gamma_{a}^{d_{a}(o)}$ (4)

Equation 4 can be interpreted as– "The more you identify with an entity in
your identity set, the less is your fairness threshold towards it". Formally,
there is a negative correlation between the fairness threshold and the extent
of identification. Next, using this equation, we plot the association-based
fairness threshold in Figure 7 for different values of transcendence level
$\gamma$. We observe that as an agent’s semantic distance with an identity
element increases, the value of $\tau$ also increases. For different values of
$\gamma$, we observed that $\gamma$ contributes to the rate of change of
$\tau$ with $d$. For increasing semantic distance $d$, $\tau$ of an agent with
a low $\gamma$ value steadily increases, whereas $\tau$ of an agent with high
$\gamma$ gradually increases. In this case, the utility of the agents is more
significantly influenced by $\gamma$ and $d$.

###  6.1 Utility Plots

We vary $\gamma$ and $d$ (and in turn $\tau$ which depends on these), to
understand its effect on the utility of the agent. Figure 8 shows the utility
plots for the agents for different values of $\gamma$. In this case, the
nature of utility curves is more diverse as compared to Section 5 for agents
with agent-based fairness threshold, because in this case, $\tau$ is also
adapting with the change in the value of $d$. This also contributes to a
larger yellow-shaded region in these utility plots denoting greater variation
in utility.

Like earlier, blue circles indicate the split at which the allocator gets the
maximum utility and the red circles indicate the split at which the recipient
gets its minimum acceptable utility. We observe that with the increase in
$\gamma$, it takes a higher $d$ for the allocator to take the whole resource
for itself. Also, as the recipient, with the increase in $\gamma$, the split
that gives the minimum acceptable utility increases slowly as the distance
increases.

###  6.2 Acceptable Splits for the Recipient

To understand more about the nature of deals that get accepted, we plot the
matrices by varying the distance of the recipient from the allocator on the
y-axis and the splits offered to it on the x-axis. The colour of cells in
matrices indicates whether the deal is accepted as green or rejected as red.
Figure 9 shows the matrices for different values of $\gamma$. We observe that
at a lower distance, the recipient accepts almost all splits. However, as
$\gamma$ increases, the agents are more tolerant and accept splits greater
than 0.5 up to low semantic distances to the allocator.

We also note in Figure 9(a) that the switch of the recipient to accept splits
greater than 0.5 happens rapidly when the $\gamma$ of the agent is low, but it
takes a higher distance for the recipient with high $\gamma$ to switch to
selfish splits (Figure 9(c)).

##  7 Conclusions

Allocation games are important to model scenarios of limited resource
allocation. We noted that just incorporating responsible behaviour in agents
is not sufficient for fair deals in this scenario. The baseline model of
transcendence was found to be insufficient and unfair in the case of the
allocation games since the agents factored in the utility of all the aspects
they identified with, without considering the variation in the absolute value
of utilities. The nature of allocation games characteristically differs from
other game-theoretic scenarios like prisoners’ dilemma, since it involves the
distribution of a fixed resource, which can be split in multiple ways and yet
the collective payoff of all the players always remains the same in all
possible game states. Thus the notion of fairness is crucial to be modelled in
agents which operate in scenarios representing the allocation games.

In this work, we focused on the Ultimatum Game (UG), which has been widely
studied to better understand human behaviour in allocation scenarios. Though
rational choice theories indicate that agents behave selfishly, empirical
evidence suggests the contrary. We extended the identity-based model of agents
- computational transcendence, to allocation games, and attempted to
understand different factors which affect the resultant game states. We
explored the interplay of the notion of fairness with subjective identity in
the context of allocation games. Also, this model can be used to simulate
agents with diverse behaviours and preferences similar to the variations
observed in people across different cultures in the context of allocation
games.

## Acknowledgement

We would like to thank Karthik Sama and Arpitha Malavalli for engaging in
insightful discussions. We would also like to thank the Machine Intelligence
and Robotics (MINRO) Center funded by the Government of Karnataka, India and
the Center for Internet of Ethical Things (CIET) funded by the Government of
Karnataka, India and the World Economic Forum for funding and supporting this
work.

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