An Unbiased Rational Decision Making Procedure for

An Unbiased Rational Decision Making Procedure for
Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence
Max-Prob: An Unbiased Rational Decision Making
Procedure for Multiple-Adversary Environments
Anat Hashavit and Shaul Markovitch
Computer Science Department, Technion—Israel Institute of Technology, Haifa 32000, Israel
{anatha, shaulm}
ago. Since then, extensive research effort has gone into developing algorithms for decision making in adversarial twoagent environments. But what happens when the number of
agents is greater than two?
Ideally, we would expect the known algorithmic solutions
for a single-adversary environment to be scalable, and fit the
more general domain of a multiple-adversary environment.
However, as we will show in the next section, the generalized minimax strategy (or its variation, the Paranoid algorithm [Sturtevant and Korf, 2000]), which assumes that
all other agents are trying to minimize the agent’s utility, is
too conservative and might result in a suboptimal decision.
While an agent using Paranoid assumes any other agent is
concerned only with its utility, the MaxN algorithm [Luckhart and Irani, 1986] takes the opposite approach and assumes
that all other agents are concerned only with their own utility.
While this strategy works in general non-cooperative environments, we will show that it also might lead to suboptimal
behavior in adversarial setups, where there are dependencies
between the agents’ utilities.
In this work we define a multiple-adversary environment
where the utility of a final state for an agent can be either
1(win) or 0(lose). We then define an agent that seeks to maximize its utility value, but is equally likely to choose between
states with the same utility value. We call such an agent an
unbiased rational agent. We show that a multiple-adversary
environment with unbiased rational agents induces a probability distribution over the possible outcomes of the game. We
claim that an unbiased rational agent acting in this environment should strive to maximize its winning probability. We
then present a new algorithm, Max-Prob, for unbiased rational lookahead-based decision making in a multiple-adversary
environment and show that existing algorithms may act suboptimally in such a setup. In addition, we present a variation
of the algorithm for a resource-bounded setup and empirically
show its advantage over alternative approaches.
In binary-utility games, an agent can have only two
possible utility values for final states, 1 (win) and
0 (lose). An adversarial binary-utility game is one
where for each final state there must be at least one
winning and one losing agent. We define an unbiased rational agent as one that seeks to maximize
its utility value, but is equally likely to choose between states with the same utility value. This induces a probability distribution over the outcomes
of the game, from which an agent can infer its probability to win. A single adversary binary game is
one where there are only two possible outcomes,
so that the winning probabilities remain binary values. In this case, the rational action for an agent is
to play minimax. In this work we focus on the more
complex, multiple-adversary environment. We propose a new algorithmic framework where agents try
to maximize their winning probabilities. We begin
by theoretically analyzing why an unbiased rational
agent should take our approach in an unbounded
environment and not that of the existing Paranoid
or MaxN algorithms. We then expand our framework to a resource-bounded environment, where
winning probabilities are estimated, and show empirical results supporting our claims.
1 Introduction
An intelligent agent is an entity that perceives its environment
and acts upon it in order to achieve an assigned goal. Which
action to take is typically decided by lookahead search in the
space of possible world states. The agent uses its perception to determine the current state of the world and evaluates
the outcome of alternative action sequences, eventually selecting a sequence it believes will end at a goal state. Adversarial agents operating in the same environment might,
however, benefit from preventing the agent from achieving
its goal. Therefore, when performing lookahead search, the
agent must consider the actions of other agents.
Many algorithms have been developed for various versions
of the above problem, mainly for a two-agent adversarial environment, the best known of which is the minimax algorithm [Shannon, 1950], suggested more than half a century
2 Problem Analysis: Unbounded Unbiased
Rational Agents
The process where multiple agents commit actions in the environment in order to achieve their assigned goal is often referred to as a game. We define a binary-utility game as one
consisting of the following elements:
The probability of an agent a to win at a state s, Pw (s, a),
is now recursively defined, under the assumption that all
agents are unbiased rational agents:
U (s, a)
Pw (s, a) =
M ax{P
∈ Succ(s)}
w (s , a)|s
Pw (s , a) ∗ |Cmax
⎩ (s)|
s ∈Cmax (s)
(a = T urn(s)
and s ∈
/ F)
Cmax (s) = arg
Figure 1: Max-Prob. Example of a complete tree.
s ∈succ(s)
Pw (s , T urn(s)).
The Max-Prob algorithm for an unbounded unbiased rational
agent, a ∈ A, computes Pw (s , a) for all successors of the
current state and selects the move leading a to a state with
maximal Pw . If several such moves exist, it selects one randomly.
A running example of the algorithm on a complete tree can
be seen in Figure 1. There are three agents playing, with
agent 1 invoking the Max-Prob procedure. The vectors at the
nodes represent the Pw values of all three agents. At node a,
agent 2 is the one to move. Since it cannot win at this state
and is indifferent as to which of the other two agents will, it
randomly selects either node c or d. Therefore, the winning
probability of agents 1 and 3 is 0.5. Similarly, the winning
probabilities for node b are 0.66, 0 and 0.66 for agents 1, 2, 3
respectively. Therefore, according to Max-Prob, agent 1 will
select the move leading it to node b, where its probability of
winning is higher.
Note that, unlike other algorithms such as MaxN, where
every value in a mid-game tree node coincides with a value
of a leaf node, our algorithm can have mid-game tree-node
values that do not coincide with any of the leaf nodes, for
example, nodes a and b.
The expected utility value of agent 1 is equal to its probability to win the game. Agent 2 is an unbiased rational agent.
It will choose node c with a probability of 0.5, and either
node e or node g with a probability of 13 . The expected utility
value of agent 1 will therefore depend on the probability of it
choosing between nodes a and b.
The generalized minimax algorithm (and Paranoid) will
propagate a value of 0 to both nodes a and b, and will then
select one randomly. This means that any tie-breaking rule
that will not strictly choose node b will cause sub-optimal
play in an environment of unbiased rational agents. For the
case of a uniformly-distributed tie-breaker, the agent will suboptimally choose node a, with probability 0.5. The expected
utility value of Paranoid in this case will be 12
MaxN will randomly select the values of one of the two
nodes, either c or d, to propagate to node a and similarly one
of e, f , or g to propagate to node b. This means that there
are six possible propagation combinations. In two of them
the agent will select b, in one it will select a, and in the other
three it will select randomly. The algorithm’s performance
will again depend on the tie-breaker. But even if the agent
will use a uniformly-distributed tie-breaker like the one that
A set of agents A = {a1 , ...., an }.
A set of game states, S.
An initial state si ∈ S.
A set F ⊆ S of final states.
A function, Turn : S → A, which designates the acting
agent in the given state. We assume that only one agent
acts at each state.
6. A function, Succ : S → 2S , which returns all the states
directly succeeding Turn(s), under the constraints of the
game, s.t ∀n, s ∈ Succ(s)n , where Succ(s)n denotes
the set of states reachable from s after a sequence of n
7. A binary utility function, U : F × A → {0, 1}, which
returns, for an agent Ai and a final state sf ∈ F , a binary
score, denoting whether the agent has won.
We further define an adversarial binary-utility game as one
where for each f ∈ F :
|A| > |{a ∈ A|U (f, a) = 0}| > 0.
This condition, that at each state there is at least one winner
and one loser, is a minimal condition for conflict between
We call a state s ∈ S that satisfies Succ(s) ⊆ F a pre-final
state. We further define for such a state the set of winning
states as W (s, a) = {s ∈ Succ(s)|U (s , a) = 1}, and similarly the set of losing states L(s, a). We define an unbiased
rational agent as one that seeks to maximize its utility value
but is equally likely to choose between states in which it has
the same utility value. In a pre-final state, such an agent selects a state from W (s, a) with equal probability if W (s, a)
is not empty, and selects one from L(s, a) with equal probability otherwise.
In this work we focus on multiple-adversary games where
n > 2. In such an environment, unbiased rational agents induce a probability distribution over the possible outcomes of
the game. This is shown in figure 1. At node a for example,
agent 2 will choose between nodes c or d with equal probability, causing the game to have two possible outcomes, each
with a probability of 0.5.
Given the above definitions, in an environment consisting
of unbiased rational agents with unbounded resources, each
agent should pick a move that maximizes its probability to
win and, if several such moves exist, to choose one randomly
with uniform distribution.
This value can also be computed by the calculation presented
for the general case.
unbiased rational agents use, it will still sub-optimally choose
a with a probability of 12
. The expected utility value of MaxN
in this case will be approximately 0.6.
Max-Prob will always select node b. Its expected utility
will therefore be 0.66, higher than that of both MaxN and
3 The Max-Prob Algorithm: Bounded Agents
It is well known that expanding the complete game tree is
computationally impossible for most if not all games. Thus,
a partial game tree is usually explored. In this section we will
show a version of the Max-Prob algorithm that is suitable for
bounded unbiased rational agents. As a bounded unbiased
rational agent usually cannot expand the search tree down to
the level of the final states (the leaves of the game tree), our
challenge is to find a way to estimate the Pw values of internal
game-tree nodes that are leaves of the search tree.
Our solution is based on traditional heuristic evaluation
functions, but with an additional requirement. We define a trait of heuristic functions for games, called windistinguishing, and restrict our algorithm to use only such
heuristic functions. A win-distinguishing heuristic function
is one that at the end of the game assigns the winners of the
game, and only them, an equal value that is higher than that
of all other agents.
Figure 2: Max-Prob. Partial tree evaluation.
The internal integral evaluates P (y ≤ x), where y is the
value associated with agent a and x is the value associated
with agent a. For a uniform distribution this value is 1 when
uha (s) < lha (s) and 0 when uha (s) < lha (s) (thus zeroing
Definition 1 (win-distinguishing) Let ai ∈ A be an agent.
A heuristic function hai : S −→ R, is win-distinguishing iff
∀s ∈ F, U (s, ai ) = 1 ⇔ hai (s) ≥ haj (s) ∀i = j.
Our goal is to use heuristic estimation to estimate Pw ,
the probability of winning the game. If we use a windistinguishing heuristic function, then the winning probability of an agent a at a state s is the probability of this agent to
have the highest heuristic value at the end of the game.
Let s ∈ S be a game state. We denote the set of all final
states in F that are reachable from s as Fs . Under the assumption of unbiased rational play defined in the previous section,
there is a probability distribution PFs over Fs that determines
for each final state f ∈ Fs the probability of reaching it under
unbiased rational play from state s. Given a heuristic function ha , we denote by fhsa the probability density function
describing the distribution of ha ’s values over the set of all
final states Fs = {f ∈ Fs |PFs (f ) > 0}.
For practical reasons, we assume for now that fhsa is uniform over an interval [lha (s), uha (s)]. In that case the winning probability of each agent is estimated by the accumulation of winning probabilities for each possible value in the
range [lha (s), uha (s)], as described in the following equation
The value for a2 can be similarly computed, and so can the
values at nodes c and d. The winning probabilities are then
propagated up the tree according to the Max-Prob algorithm.
At node e agent 2 has only one maximal child, but for node f
both children maximize the agent’s winning probability and
so the expected probability values propagated for agents 1 and
3 are not identical to either of their values at the leaf nodes.
Pˆw (s, a) =
ha (s)
fhsa (x)·
x=lha (s)
a ∈A\{a}
h (s)
the whole expression); otherwise it is u (s)−l
Considering the above observations, we suggest an adaption of the algorithm presented in the previous section. The
algorithm will remain the same except for the leaf evaluation
method, which will first produce a vector of intervals bounding the heuristic value for each agent, and then infer from
these intervals the probability of each agent to win according to Equation 1. The values propagated up the tree will be
the estimated winning probabilities. The algorithm is listed
in Figure 3.
An example of how Max-Prob assigns values in a partial
game tree is presented in figure 2. At node (a) the value of
agent 1 will always be higher than that of all the other agents.
Therefore, it will have a winning probability of 1 while all
the other agents will have a winning probability of 0. Each of
the other leaf nodes contain intervals that intersect with each
other. We will need to compute the winning probabilities,
according to the rules of uniform distribution and Bayes’ theorem. At node b, for example, agent number 3 has no chance
of being the winner and is quickly taken out of the equation.
For the other two agents we will need to examine the possible
heuristic values of the final states in Fs . For a state f where
ha2 (f ) ∈ [3, 4], a1 has probability of 1 to win, and again
for the case where ha2 (f ) ∈ [4, 5], and ha1 (f ) ∈ [5, 6]. If
ha2 (f ) ∈ [4, 5] and ha1 (f ) ∈ [4, 5], a1 has probability of 0.5
to win. The probabilities of each of these scenarios to occur
are 0.5, 0.25, 0.25 respectively, and so
Pˆw (b, a1 ) = 0.5 ∗ 1 + 0.25 ∗ 1 + 0.25 ∗ 0.5 = 0.875.
fhsa (y)dy dx.
4 Empirical Evaluation
Ideally, we would estimate the probability that ha is maximal
for an arbitrary leaf f ∈ Fs . Since this probability is too difficult to
estimate, we instead estimate the win probability by comparing the
expected h values (or their bounds) for the agents.
Experimental Methodology
We compared the Max-Prob algorithm with MaxN and Paranoid. In addition, we compared it to the recently developed
Chinese Checkers
Tournament Spades 8
Tournament Hearts 8
Average win percentage
Procedure M AX -P ROB(s, a)
C ← Succ(s)
Pw ← M ax{WinProb(c)[a] | c ∈ C}
Cmax ← {c ∈ C|WinProb(c)[a] = Pw (s)}
Return rand(Cmax )
Procedure W IN P ROB(s)
If s ∈ F
Foreach a ∈ A
Pw [a] = Pˆw (s, a)
Return Pw
C ← Succ(s)
curr = T urn(s)
Pw [curr] ← M ax{WinProb(c)[curr]|c ∈ C}
Cmax ← {c|c ∈ C, WinProb(c)[curr] = Pw [curr]}
Foreach a ∈ A\curr
Pw [a] ←
|Cmax | Pw (c)[a]
Return Pw
Table 1: Winning percentage for each game
the game is to have the highest score among all players. In the original version of the game, the scoring is
affected by the number of tricks a player took and the
bid she placed at a preliminary phase. Our simplified
version omits the bidding phase and awards 10 points
for every trick taken. The heuristic function we used
was constructed from the number of tricks taken and the
composition of the player’s hand.
2. Perfect-information Hearts Hearts is another hidden information trick taking card game. Each heart card taken
is worth 1 penalty point and the queen of spades is worth
13. The goal of the game is to have the lowest number
of penalty points at the end of the game. Our heuristic
function consisted of the number of penalty points taken
by the players and the composition of the players’ hand.
3. ChineseCheckers Chinese Checkers is an abstract strategy game that can be played with up to six players. The
goal of the game is to be the first player who moves all
ten of her pawns from her home camp, located at one pit
of a hexagram, to her destination camp, located at the
opposite pit. Our heuristic was simple and consisted of
the accumulated distance of the pawns from their destination camp.
4. Abalone Abalone is another abstract strategy game. Its
board is an hexagon containing 61 slots. In the three
player version which we play, each player starts the
game with eleven marbles arranged in two adjunct rows,
one containing 5 marbles and another containing 6. The
goal of the game is to be the first player who manages
to push six of her opponents’ marbles out of the game
board. The heuristic function we used contained the
number of opponents’ pawns pushed off the board and
the pawns’ arrangement on the board.
Each of our heuristic functions can be bounded by a constant
number. We used these numbers as the upper bounds for the
Max-Prob intervals and the current heuristic value of a state
as the lower bound. Tie-breaking was done only at the root
level, where we preferred states with a higher heuristic value.
For the MP-Mix algorithm we set the thresholds to values that
indicate a notable improvement in an agent’s status: a trick
taken for Spades, a penalty point for Hearts, pushing another
agent’s pawn off the board for Abalone, and a three slots distance gap for Chinese Checkers.
Figure 3: Max-Prob. Algorithm description.
MP-Mix algorithm [Zuckerman et al., 2009], which alternates
between playing MaxN, Paranoid, and an offensive strategy
that attacks the leading agent. It does so by examining the
difference between the heuristic value of the leading agent
and that of the runner up. If the agent is the leading agent
and the said difference is higher than a predefined threshold,
Td , a paranoid strategy is taken. Otherwise, if the leading
agent is another agent and this difference is higher than a second threshold, To , an offensive strategy is taken. In all other
cases, the MaxN strategy is played.
The experiments were conducted on four different games,
two board games and two card games. For the card games
setup, we used the perfect information version of the games.
The depth of the search tree was limited to 8 plies, and we
did not allow the search to end in the middle of a trick. Algorithm performance was compared on 100 different hands that
were played for each of the 24 possible player orderings using two different setups. In the first setup, we ran 100 single
hand games for each player ordering, and in the second a single 100 hands game for each player ordering. An agent was
rewarded with one victory point for each game where it outperformed its opponents. Since these are not single winner
games, several agents could gain a victory point in a single
The board games have a much higher branching factor than
the card games and so their setup was slightly different. Each
game tested three of the four competing algorithms, and the
depth of the tree was limited to 4.
The games we used are:
1. Perfect-information Simplified Spades Spades is a trick
taking card game for which the spades suite is always
a trump. The game has both a partnership and a solo
version; we will focus on the solo version. The goal of
Table 1 presents the percentage of games won by each player.
It can be seen that the Max-Prob algorithm outperforms its
opponents with an average win percentage that is more than
20% higher than that of the runner up Paranoid, which had
generalizes the classical minimax algorithm from a twoplayer game into an N-players game by assuming that a coalition of N-1 players has been formed to play against the remaining player. It is ”paranoid” in the sense that it assumes
collaboration against a single player. Unlike algorithms that
preserve the N player structure of the game, Paranoid can
use deep pruning, as shown by Korf [1991]. This is a distinct
advantage of the algorithm. However, the paranoid assumption can lead to suboptimal play [Sturtevant and Korf, 2000].
Furthermore, using Paranoid renders all individual opponent
modelings unusable, even if these are available.
Zuckerman et al. [2009] presented the MP-Mix algorithm,
which implements a mixed strategy approach. When it is an
agent’s turn to act, it examines the difference between the
leading agent’s heuristic value and that of the runner up. It
then decides accordingly whether to use an offensive strategy, a paranoid strategy, or the default MaxN strategy. This
algorithm overcomes MaxN’s inherent problem of the agents’
indifference to each other by explicitly examining the relation between the leading agent and the runner up. However,
this solves the indifference problem only in part since it does
not take into account the status of the other agents, whereas
Max-Prob does so implicitly in the calculation of the winning
Probabilistic methods, and methods which use ranges of
values, have been used mainly as a tool for better incorporating opponent models in the search algorithm, or for the
purpose of selective deepening. Most such methods were implemented only in single-adversary environments. The B ∗
algorithm proposed by Berliner [1979] defines for each node
an interval, bounded by an optimistic and a pessimistic value.
These intervals are then used to selectively expand nodes.
The interval values of a node are derived from its children’s
values and are backed up the tree, in an attempt to narrow the
range of the root’s interval until it is reduced to a single value.
Palay [1985] extended the B ∗ algorithm to a version where
the evaluation of a leaf produces probability distributions,
which are then propagated up the tree using the product propagation rule. Baum and Smith [1997] proposed a Bayesian
framework to selectively grow the game tree and evaluate tree
nodes. Their algorithm, called BP, describes how to evaluate the uncertainty regarding the evaluation function. Instead of assigning single values to nodes, discrete distributions on the possible values of the leaves are assigned and
propagated up the tree, again using the product propagation
rule. These algorithms are similar to the Max-Prob algorithm
in that the leaf evaluation procedure does not produce a single
value. But they continue to propagate nondiscrete values, unlike Max-Prob, which propagates vectors of discrete winning
probabilities deduced from the evaluation of the leaves.
The M∗ algorithm [Carmel and Markovitch, 1996] deals
with an agent that is uncertain as to the evaluation function
of its opponents and thus tries to contain it within an error
bound, similar to the Max-Prob algorithm. Unlike the MaxProb algorithm, however, M∗ does not convert these intervals to single values but rather propagates intervals of values.
Donkers [2001] proposes PrOM, an extension to opponent
modeling search that takes into account uncertainties in opponent models and allows several models to be considered,
better results than both MaxN and Mp-Mix. It would appear
that, given no other options, playing according to a paranoid
assumption will bring better results than playing according to
the indifference assumption of MaxN.
Although Td and To were set to values which reflected
a genuine advantage in the games used for testing, Mp-Mix
ranked last in most of the domains tested. These results are
inconsistent with the results presented in the original paper.
Therefore, except for the need to perform a parameter tuning process before competing Mp-Mix with other algorithms,
we cannot draw any conclusions regarding its performance in
relation to Max-Prob. We do believe that with additional parameter tuning, for each of the domains tested, Mp-Mix can
achieve much better results.
For the board games, we did a lot better in Chinese Checkers than in Abalone. In the card games, Max-Prob achieved
the best results for tournaments, even though scoring of previous games was taken into account in the heuristic function,
which was equal for all agents.
The goal of each of the games we tested was to minimize
or maximize some metrics in respect to the other agents. The
heuristic functions were constructed from two components, a
direct one, which contained the current value of the metrics,
and an indirect one, which attempted to estimate any future
change in metric values. We can see that Max-Prob performs
better in setups where the indirect part has less weight.
In Chinese Checkers, where Max-Prob wins more than
half the games, we minimize the accumulated distance of the
pawns from their home camp, and this metric is the only component of the heuristic function. In the card games we used
heuristic functions that contained both a direct and an indirect component. The direct component counted the number
of tricks or penalty points of a player and the indirect component used the composition of the player’s hand in order to
estimate future changes to these metrics. Max-Prob had less
of an advantage in the single-hand card games than in Chinese Checkers. In the tournament setups, however, the performance improved notably. This is due to the diminishing
effect of the indirect parts of the heuristics as more games
are played. In Abalone, an opponent’s pawn is pushed off
the board in a small percentage of the moves, and so the indirect component of the heuristic was the one affecting the
win probability calculation for most of the game. This made
it difficult for Max-Prob to accurately estimate the winning
probability, as reflected by the results.
5 Related Work
Luckhart and Irani [1986] presented the MaxN algorithm. In
this algorithm, evaluation of a leaf in the game tree produces
a vector of N values. Each component in the vector is the
utility value of the corresponding player for that state. The
algorithm also defines a propagation rule which states that
each agent in its turn selects a move maximizing it individual utility value, without considering the utility values of the
other players. This means that there is an unstated assumption
that the agents are indifferent to each other’s performance, an
assumption that does not hold for adversarial games.
Sturtevant [2000] proposed the Paranoid algorithm which
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6 Conclusions
We presented a new framework for lookahead-based decision
making in a multiple-adversary environment. Our framework
defined unbiased rational agents and the way such agents
should act for both bounded and unbounded resource environments. We validated our framework by theoretical analysis and empirical evaluations.
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